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Trends in Mathematics

Piotr Kielanowski Anatol Odzijewicz Emma Previato Editors

Geometric Methods in Physics XXXV Workshop and Summer School, Białowieża, Poland, June 26 – July 2, 2016

Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference.

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Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English.

Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome.

We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser.

Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference.

More information about this series at http://www.springer.com/series/4961

Piotr Kielanowski • Anatol Odzijewicz Emma Previato Editors

Geometric Methods in Physics XXXV Workshop and Summer School, Białowieża, Poland, June 26 – July 2, 2016

Editors Piotr Kielanowski Departamento de Física CINVESTAV Ciudad de México, Mexico

Anatol Odzijewicz Institute of Mathematics University of Białystok Białystok, Poland

Emma Previato Department of Mathematics and Statistics Boston University Boston, MA, USA

ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-319-63593-4 ISBN 978-3-319-63594-1 (eBook) https://doi.org/10.1007/978-3-319-63594-1 Library of Congress Control Number: 2017962618 Mathmatics Subject Classification (2010): 01-06, 01A70, 20N99, 58A50, 58Z05, 81P16, 33D80, 51P05 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

G.A. Goldin In Memory of S. Twareque Ali . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Part I: Geometry & Physics A. Antonevich Quasi-periodic Algebras and Their Physical Automorphisms . . . . . . . . .

3

I. Beltiţă, D. Beltiţă and B. Cahen Berezin Symbols on Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

¸ T. Brzeziński and L. Dabrowski A Curious Differential Calculus on the Quantum Disc and Cones . . . .

19

M. Fecko Nambu Mechanics: Symmetries and Conserved Quantities . . . . . . . . . . .

27

L. Jeffrey, S. Rayan, G. Seal, P. Selick and J. Weitsman The Triple Reduced Product and Hamiltonian Flows . . . . . . . . . . . . . . . .

35

B. Mielnik The Puzzle of Empty Bottle in Quantum Theories . . . . . . . . . . . . . . . . . .

51

V.F. Molchanov Poisson Transforms for Tensor Products in Compact Picture . . . . . . . .

61

T. Kobayashi, T. Kubo and M. Pevzner Conformal Symmetry Breaking Operators for Anti-de Sitter Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

A.K. Prykarpatski Hamilton Operators and Related Integrable Differential Algebraic Novikov–Leibniz Type Structures . . . . . . . . . . . . . . . . . . . . . . . . .

87

vi

Contents

C. Roger An Algebraic Background for Hierarchies of PDE in Dimension (2|1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

A.I. Shafarevich Lagrangian Manifolds and Maslov Indices Corresponding to the Spectral Series of the Schrödinger Operators with Delta-potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

J. Smotlacha and R. Pincak Electronic Properties of Graphene Nanoribbons in a Uniform Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 E. Stróżyna Formal Normal Forms for Germs of Vector Fields with Quadratic Leading Part. The Rational First Integral Case . . . . . . . . . . . . . . . . . . . . .

119

H. Żo ladek ¸ The Poncelet Theorems in Interpretation of Rafa l Ko lodziej . . . . . . . . . 129

Z. Pasternak-Winiarski and T.L. Żynda Weighted Szegő Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Part II: Integrability & Geometry I. Cheltsov On a Conjecture of Hong and Won . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 C.D. Guevara and S.P. Shipman Short-time Behavior of the Exciton-polariton Equations . . . . . . . . . . . . . 175 I. Horozov Periods of Mixed Tate Motives over Real Quadratic Number Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 W.X. Ma and X. Lü Soliton Hierarchies from Matrix Loop Algebras . . . . . . . . . . . . . . . . . . . . . . 191 C.A. Tracy and H. Widom On the Ground State Energy of the Delta-function Fermi Gas II: Further Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 D. Zakharov and V. Zakharov Non-periodic One-gap Potentials in Quantum Mechanics . . . . . . . . . . . . 213

Contents

vii

Part III: Abstracts of the Lectures at “School on Geometry and Physics” A. Bolsinov Integrable Geodesic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229

M. Bożejko Anyonic Fock Spaces, q-CCR Relations for |q| = 1 and Relations with Yang–Baxter Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 T. Brzeziński Differential and Integral Forms on Non-commutative Algebras . . . . . . . 249 J. Kijowski General Relativity Theory and Its Canonical Structure . . . . . . . . . . . . . . 255 B. Mielnik Exponential Formulae in Quantum Theories . . . . . . . . . . . . . . . . . . . . . . . . . 261 E. Previato Complex Algebraic Geometry Applied to Integrable Dynamics: Concrete Examples and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Th. Voronov Volumes of Classical Supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Trends in Mathematics, ix–x c 2018 Springer International Publishing

Preface This book contains a selection of papers presented during the Thirty-Fifth “Workshop on Geometric Methods in Physics” (WGMPXXXV) and abstracts of lectures given during the Fifth “School on Geometry and Physics”, which both took place in Bia lowieża, Poland, in the summer 2016. These two coordinated activities are an annual event. Information on the previous and the upcoming occurrences and related materials can be found at the URL: http://wgmp.uwb.edu.pl.

The volume is divided into four parts. It opens with a paper dedicated to the memory of S. Twareque Ali – for many years an active member of the Organizing Committee of our workshop who died suddenly in 2016. The second part, “Geometry and Physics”, includes papers based on talks delivered during the workshop.

The third part, “Integrability and Geometry”, is based on the eponymous special session, organized by G.A. Goldin, A. Odesskii, E. Previato, E. Shemyakova and Th. Voronov. The final part contains extended abstracts of the lecture-series given during the Fifth “School on Geometry and Physics”.

The WGMP is an international conference organized each year by the Department of Mathematical Physics in the Faculty of Mathematics and Computer Science of the University of Bia lystok, Poland. The main theme of the workshops, consistent with the title, is the application of geometric methods in mathematical physics and it includes a study of non-commutative systems, Poisson geometry, completely integrable systems, quantization, infinite-dimensional groups, supergroups and supersymmetry, quantum groups, Lie groupoids and algebroids as well as related topics. Participation in the workshops is open; the typical audience consists of physicists and mathematicians from many countries in several continents with a wide spectrum of interests.

Workshop and School are held in Bia lowieża, a village located in the east of Poland near the border with Belarus. Bia lowieża is situated on the edge of the Bia lowieża Forest, shared between Poland and Belarus, which is one of the last remnants of the primeval forest that covered the European Plain before human settlement and was designated a UNESCO World Heritage Site. The peaceful atmosphere of a small village, combined with natural beauty, yields a unique environment for learning and cooperating: as a result, the core audience of the WGMPs has become a strong scientific community, documented by this series of Proceedings.

The Organizing Committee of the 2016 WGMP gratefully acknowledges the financial support of the University of Bia lystok and the Belgian Science Policy Of-

x

Preface

fice (BELSPO), IAP Grant P7/18 DYGEST. Thanks also go to the U.S. National Science Foundation for providing support to participants in the “Integrability and Geometry” session of the event, Grant DMS 1609812. Last but not least, credit is due to early-career scholars and students from the University of Bia lystok, who contributed limitless time and effort to setting up and hosting the event, aside from being active participants in the scientific activities.

The Editors

Participants of the XXXV WGMP (Photo by Tomasz Goliński)

Geometric Methods in Physics. XXXV Workshop 2016 Trends in Mathematics, xi–xvi c 2018 Springer International Publishing

In Memory of S. Twareque Ali Gerald A. Goldin Abstract. We remember a valued colleague and dear friend, S. Twareque Ali, who passed away unexpectedly in January 2016.

S. Twareque Ali in Bia lowieża.

xii

G.A. Goldin

1. Remembering Twareque Syed Twareque Ali, whom we all knew as Twareque, was born in 1942, and died in January 2016. This brief tribute is the second one I have prepared for him in a short period of time. With each sentence I reflect again on his extraordinary personality, his remarkable career – and, of course, on the profound influence he had in my life. Twareque was more than a colleague – he was a close friend, a confidant, and a teacher in the deepest sense.

When I remember Twareque, the first thing that comes to mind is his laughter. He found humor in his early changes of nationality: born in the British Empire, a subject of George VI, Emperor of India, he lived in pre-independence India, became a citizen of Pakistan, and then of Bangladesh – all without moving from home. Eventually he became a Canadian citizen, residing with his family in Montreal for many years.

Twareque’s laughter was a balm. In times of sadness or disappointment, he was a source of optimism to all around him. His positive view of life was rooted in deep, almost unconsciously-held wisdom. Although he personally experienced profound nostalgia for those lost to him, he knew how to live with joy. He could laugh at himself, never taking difficulties too seriously.

And he loved to tell silly, inappropriate jokes – which, of course, cannot be repeated publicly. He introduced me to the clever novels by David Lodge, Changing Places, and Small World, which satirize the academic world mercilessly. In Lodge’s characters, Twareque and I saw plenty of similarities to academic researchers we both knew in real life – especially, to ourselves.

Twareque was fluent in several languages, a true “citizen of the world.” He loved poetry, reciting lengthy passages from memory in English, German, Italian, or Bengali. In Omar Khayyam’s Rubaiyat, translated by Edward Fitzgerald, he found verses that spoke to him. These are among them: ...

Come, fill the Cup, and in the Fire of Spring The Winter Garment of Repentance fling: The Bird of Time has but a little way To fly – and Lo! the Bird is on the Wing. ...

A Book of Verses underneath the Bough, A Jug of Wine, a Loaf of Bread – and Thou Beside me singing in the Wilderness Oh, Wilderness were Paradise enow! ...

The Moving Finger writes, and, having writ, Moves on; nor all your Piety nor Wit Shall lure it back to cancel half a Line, Nor all your Tears wash out a Word of it.

In Memory of S. Twareque Ali

xiii

2. A short scientific biography Twareque obtained his M.Sc. in 1966 in Dhaka (which is now in Bangladesh). He received his Ph.D. from the University of Rochester, New York, USA, in 1973, where he studied with Gérard Emch. Professor Emch remained an inspiration to him for the rest of his life, and Twareque expressed his continuing gratitude. In 2007, together with Kalyan Sinha, he edited a volume in honor of Emch’s 70th birthday [1]; and in 2015, he organized a memorial session for Emch at the 34th Workshop on Geometric Methods in Physics in Bia lowieża.

After earning his doctorate, Twareque held several research positions: at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy; at the University of Toronto and at the University of Prince Edward Island in Canada; and at the Technical University of Clausthal, Germany in the Arnold Sommerfeld Institute for Mathematical Physics with H.-D. Doebner. He joined the mathematics faculty of Concordia University in Montreal as an assistant professor in 1981, becoming an associate professor in 1983 and a full professor in 1990.

During his career as a mathematical physicist, Twareque achieved wide recognition for his scientific achievements. He was known for his studies of quantization methods, coherent states and symmetries, and wavelet analysis. A short account cannot do justice to his accomplishments; the reader is referred for more detail to two published obituaries from which I have drawn [2, 3], and asked to forgive the many omissions. I cannot do better than to quote the summary in another tribute I wrote [4]: “During the 1980s, Twareque worked on measurement problems in phase space, and on stochastic, Galilean, and Einsteinian quantum mechanics [5,6] Then he began to study coherent states for the Galilei and Poincaré groups, and collaborated with Stephan de Bièvre on quantization on homogeneous spaces for semidirect product groups.

“There followed his extensive, long-term, and indeed famous collaboration with Jean-Pierre Antoine and Jean Pierre Gazeau, focusing on square integrable group representations, continuous frames in Hilbert space, coherent states, and wavelets. Their joint work culminated in publication of the second edition of their book in 2014 – a veritable treasure trove of mathematical and physical ideas [7–10].

“Twareque’s work on quantization methods and their meaning is exemplified by the important review he wrote with M. Englis̆ [11], and his work on reproducing kernel methods with F. Bagarello and Gazeau [12].”

Twareque’s contributions of time and effort helped bring a number of scientific conference series to international prominence. Foremost among these was the Workshop on Geometric Methods in Physics (WGMP) in Bia lowieża (organized

by Anatol Odzijewicz). Twareque attended virtually every meeting from 1991 to 2015, where we would see each other each summer. He was a long-time member of the local organizing committee, and co-edited the Proceedings volumes. Other conference series to which he contributed generously of his energy included the University of Havana International Workshops in Cuba (organized by Reinaldo Rodriguez

xiv

G.A. Goldin

Ramos), and the Contemporary Problems in Mathematical Physics (Copromaph) series in Cotonou, Benin (organized by M. Norbert Hounkonnou).

He was also an active member of the Standing Committee of the International Colloquium on Group Theoretical Methods in Physics (ICGTMP) series.

Twareque and his wife Fauzia came together to the 29th meeting of ICGTMP in Tianjin, China in 2012. She attended the special session where Twareque (to his surprise) was honored on the occasion of his 70th birthday. Their son Nabeel, of whom he always spoke with great pride, practices pediatric medicine in Montreal.

Twareque was a deep thinker, who sought transcendence through ideas and imagination. The truths of science and the elegance of mathematics in the quantum domain were part of the mysterious beauty for which he longed – a longing shared by many great scientists, a longing that we, too, share.

S. Twareque Ali in thought at WGMP XXXIII, July 2, 2014. Photograph by G.A. Goldin.

As profoundly as Twareque cared about understanding the meanings of scientific ideas, he cared equally about inspiring his students to succeed. He helped them with personal as well as professional issues. As Anna Krasowska and Renata Deptula, two of his more recent students who came from Poland to work with

In Memory of S. Twareque Ali

xv

him, wrote [2], “If anything in our lives became too complicated it was a clear sign we needed to talk to Dr. Ali. Every meeting with him provided a big dose of encouragement and new energy, never accompanied with any criticism or judgment.”

This was Twareque’s gift – to understand, to inspire, to give of himself.

Twareque died suddenly and unexpectedly January 24, 2016 in Malaysia, after participating actively in the 8th Expository Quantum Lecture Series (EqualS8) – indeed, doing the kind of thing he loved most.

3. Concluding thoughts Twareque believed passionately in world peace, in service to humanity, and in international cooperation. He understood the broad sweep of history. His tradition was Islam, as mine is Judaism, and although neither of us adhered to all the rituals of our traditions, we shared an interest in their history, their commonalities, and their contributions to world culture. We even researched correspondences between the roots of words in Arabic and Hebrew. On a first visit to Israel for a conference in 1993, we visited Jerusalem together. Twareque did much to aid the less privileged and less fortunate – in the best of our traditions, often anonymously.

Often one closes a retrospective on someone’s life with a sunset, marking the ending of day and the beginning of night. My choice for Twareque is different. He is someone who joined a scientific mind with a spiritual heart, and for Twareque, the park and the forest in Bia lowieża were at the center of his spirituality. So I imagine him looking at us, even now, and marveling at the beauty of heavenly clouds reflected in the water.

Reflection of the heavens in Bia lowieża Park, July 4, 2013.

Photograph by G.A. Goldin.

xvi

G.A. Goldin

Acknowledgment I am deeply indebted to Twareque’s family, friends, students, and colleagues.

Thanks to the organizers of the 35th Workshop on Geometric Methods in Physics for this opportunity to honor and remember him.

References [1] S. Twareque Ali, Kalyan B. Sinha (eds.), A Tribute to Gérard G. Emch. Hindustan Book Agency, New Delhi, 2007.

[2] J.-P. Antoine, J.P. Gazeau, G. Goldin, J. Harnad, M. Ismail, A. Krasowska, R. Deptula, R.R. Ramos. In memoriam, S. Twareque Ali (1942–2016). Le Bulletin, Centre de Recherches Mathématiques 22 (1) (2016), 9–12.

[3] F. Schroeck, J.-P. Antoine, G. Goldin, R. Benaduci. Syed Twareque Ali: Obituary.

Journal of Geometry and Symmetry in Physics 41 (2016), 105–111.

[4] G.A. Goldin, In Memoriam: Syed Twareque Ali. Proceedings of the 31st International Colloquium on Group Theoretical Methods in Physics. Springer, 2017 (to be published).

[5] S.T. Ali, Stochastic localization, quantum mechanics on phase space and quantum space-time. La Rivista del Nuovo Cimento 8 (11) (1985), 1.

[6] S.T. Ali, J.A. Brooke, P. Busch, R. Gagnon, F. Schroeck, Current conservation as a geometric property of space-time. Canadian J Phys 66 (3) (1988), 238–44.

[7] S.T. Ali, J.-P. Antoine, J.P. Gazeau, Square integrability of group representations on homogeneous spaces I. Reproducing triples and frames. Ann. Inst. H. Poincaré 55 (1991), 829–856.

[8] S.T. Ali, J.-P. Antoine, J.P. Gazeau, Continuous frames in Hilbert space. Annals of Physics 222, 1–37; Relativistic quantum frames, Ann. of Phys. 222 (1993), 38–88.

[9] S.T. Ali, J.-P. Antoine, J.P. Gazeau, U.A. Müller, Coherent states and their generalizations: A mathematical overview. Reviews in Math. Phys. 7 (1995), 1013–1104.

[10] S.T. Ali, J.-P. Antoine, J.P. Gazeau Coherent States, Wavelets and their Generalizations (and references therein). Springer, 2000; 2nd edn., 2014.

[11] S.T. Ali, M. Englis̆, Quantization Methods: A Guide for Physicists and Analysts.

Rev. Math. Phys. 17 (2005), 391–490.

[12] S.T. Ali, F. Bagarello, J.P. Gazeau, Quantizations from reproducing kernel spaces.

Ann. Phys. 332 (2013), 127–142.

Gerald A. Goldin Rutgers University New Brunswick, NJ 08901, USA e-mail: geraldgoldin@dimacs.rutgers.edu

Part I Geometry & Physics

Geometric Methods in Physics. XXXV Workshop 2016 Trends in Mathematics, 3–9 c 2018 Springer International Publishing

Quasi-periodic Algebras and Their Physical Automorphisms A. Antonevich and A. Glaz Abstract. An automorphism of a quasi-periodic algebra on Rm is said to be physical, if it is generated by a mappings of Rm . The aim of this work is to give a description of the mappings, corresponding to such automorphisms.

Mathematics Subject Classification (2010). Primary 46J10; Secondary 42A75.

Keywords. Quasi-periodic function, maximal ideal space, automorphism, algebraic unit.

1. Introduction: Invariant subalgebras Invariant algebra is an important object in different fields of analysis In this paper we consider quasi-periodic algebras on Rm invariant under mappings of Rm . Quasiperiodic functions and algebras arise naturally in many fields of analysis. A list of their applications are given, for example, in [1, 2]. Among them let us single out integrating of Hamiltonian systems and nonlinear equations, the theory of conductivity and the theory of quasi-crystals.

Let B(X) be Banach algebra of all bounded functions on X equipped with sup-norm. Any mapping α : X → X generates the composition operator W a(x) = a(α(x)),

(1)

acting on B(X). The operator W is linear and multiplicative, i.e., it is an endomorphism of B(X). If α is invertible, then W is an automorphism of B(X).

A closed subalgebra A ⊂ B(X) is said to be invariant with respect to α (shortly α-invariant ) if W (A) ⊂ A. Then the operator W is an endomorphism of A. If W is invertible on A, then it is an automorphism. In this case the algebra A is called two-sided invariant.

For any given subalgebra A0 ⊂ B(X) there exists the smallest invariant closed algebra A+ containing A0 and there exists the smallest two-sided invariant closed algebra A containing A0 .

4

A. Antonevich and A. Glaz

Among the motivations to construct invariant algebras can be pointed out the following.

1.1. Investigation of weighted composition operators A weighted composition operator on B(X) is an operator of the form Bu(x) = a(x)u(α(x)),

(2)

where the coefficient a ∈ B(X) is a given function.

According to Gelfand–Naimark theorem any commutative C ∗ -algebra A with unity element is isomorphic to the algebra C(M(A)) of all continuous functions on a compact space M(A). This space is called maximal ideal space of the algebra A. The isomorphism A a→ a ∈ C(M(A)) is called Gelfand transform.

If A ⊂ B(X) is an α-invariant C ∗ -subalgebra, then endomorphism W induces a continuous mapping α : M(A) → M(A).

Proposition 1 ([3]). Let A ⊂ B(X) be an invariant C ∗ -subalgebra and a ∈ A. For the spectral radius R(B) of the weighted composition operator (2) the following variational principle holds

ln | a|dν , R(B) = max exp ν∈Λα

M(A)

where Λα is the set of all α -invariant normalized Borel measures on M(A).

Let us consider operator B of the form (2) such that a ∈ A0 , where A0 ⊂ B(X) is a C ∗ -subalgebra. In order to apply the variational principle we need to find an α-invariant algebra A containing A0 .

Example. Let X = R, α(x) = qx, q ∈ R and Bu(x) = a(x)u(qx),

(3)

where a0 is a continuous periodic function with period 1. In this case we need to construct the smallest algebra, containing all periodic functions with period 1 and invariant with respect to α(x) = qx.

1.2. Cross-product construction Let A be a C ∗ -algebra and τ : A → A be an automorphism. There exists a set of C ∗ -algebras B such that 1. A ⊂ B; 2. there exists a unitary element T ∈ B, such that τ (a) = T −1 aT ; 3. the algebra B is generated by A and T. The largest among such algebras, denoted by A ×τ Z, is called cross-product of A and its automorphism τ . A canonical construction of the cross-product was proposed by von Neumann.

Quasi-periodic Algebras

5

There exist a number of generalizations of cross-product construction to the case of endomorphism τ : A0 → A0 [4,5]. One of them is based on the following. If we construct by given algebra A0 a larger algebra A such that τ can be extended to automorphism of A, then cross-product construction is reduced to the classical case of automorphism.

2. Almost periodic algebras 2.1. Quasi-periodic algebras Let CB(Rm ) be the space of all bounded continuous functions on Rm . The smallest closed subspace in CB(Rm ) containing all functions ei2π<h,x> , x ∈ Rm , h ∈ Rm is the algebra CAP (Rm ) of continuous almost periodic functions [6].

Any C ∗ -subalgebra of A ⊂ CAP (Rm ) is called almost periodic. A closed subalgebra A ⊂ CB(Rm ) is called quasi-periodic, if it is generated by a finite number of functions ei2π ±hj ,x , hj ∈ Rm , j = 1, 2, . . . , N. To any almost periodic function a corresponds formal Fourier series a(x) ∼

∞

Cj ei2π ξj ,x .

j=1

The vectors ξj are called frequencies of the function, the set {ξj } is called the spectrum of the function a.

For a given almost periodic algebra A denote by H(A) the union of spectra of all functions from A. The set H(A) ⊂ Rm is a subgroup in Rm and is called the frequencies group of the algebra A. The subgroup Γ ∈ Rm with a finite number of generators is called the quasilattice. As an abstract group, any quasi-lattice Γ is isomorphic to ZN , where N is the number of independent generators.

In this terminology a subalgebra A is quasi-periodic, if H(A) is a quasi-lattice.

If H(A) ≈ ZN then A is called the algebra with N quasi-periods.

2.2. Gelfand transform of almost periodic algebras Let G be a commutative locally compact group. Any continuous homomorphism f from G into the unite circle S1 = {z ∈ C : |z| = 1} is called the character of which is also locally group G. The set of all characters forms the dual group G, compact.

According to the Pontryagin duality [7], if G is discrete, then the dual group is compact.

G

6

A. Antonevich and A. Glaz

Theorem 2. Let A be a C ∗ -subalgebra of CAP (Rm ). Then

M(A) = H(A), i.e., the space of maximal ideals is the dual group to the frequencies group.

Consider the following examples.

1. Rm can be considered as a discrete group. It is a group with an uncountable m is called Bohr compact and does not

set of generators. The dual group R have an explicit description.

Group Rm is the frequencies group of the algebra CAP (Rm ) of all almost periodic functions. Therefore the space of maximal ideals of the algebra CAP (Rm ) is the Bohr compact.

2. If A is a quasi-periodic algebra, then H(A) = ZN for some N and the dual N = TN . It follows that the Gelfand transgroup is a N -dimensional torus: Z form gives an isomorphism A → C(TN ) ∼ CP (RN ), where CP (RN ) is the space of continuous function on RN periodic with period 1 for each variable.

The isomorphism CP (RN ) → A (inverse to Gelfand transform) can be constructed as follows.

Let us consider a linear embedding Rm → L ⊂ RN , where L is an m-dimensional vector subspace. Then the restrictions of functions from CP (RN ) on the L form a quasi-periodic algebra AL on Rm whose frequencies group is the orthogonal projection of the lattice ZN onto L. A subspace L ⊂ RN is said to be totally irrational, if there are no vectors from ZN that are orthogonal to L (except zero vector). If the subspace L is totally irrational, then H(AL ) ≈ ZN and AL is a quasi-periodic algebra with N generators.

Using different totally irrational embedding of Rm into RN there can be obtained any quasi-periodic algebra on Rm with N quasi-periods. These algebras are isomorphic to each other as abstract algebras, but differently realized as subalgebras of CAP (Rm ).

Like in the paper [1] the space Rm will be called as the physical space and the space RN as the super-space.

3. Automorphisms of quasi-periodic algebra 3.1. Statement of the problem Let A be a quasi-periodic algebra on Rm with N quasi-periods. Each automorphism τ : A → A is generated by a homeomorphism τ of the torus TN (and by the corresponding covering mapping τ of the super-space RN ).

An automorphism τ of A is called physical, if it is generated by a mapping α : Rm → Rm of the physical space. A general problem is to give a description

Quasi-periodic Algebras

7

of all physical automorphisms of the quasi-periodic algebras. We remark that the symmetry group of quasi-crystal consists on such mapping and the problem under consideration is connected with investigation of quasi-crystallographic groups [2].

Here we consider the following case of general problem.

Let A0 be a given quasi-periodic algebra on Rm and α : Rm → Rm be a given mapping. In general, algebra A0 may be not invariant under α and the smallest invariant (two-sided invariant) algebra A containing A0 can be not quasi-periodic.

The question is: for which mapping α of the physical space Rm the smallest invariant (two-sided invariant) algebra containing A0 is quasi-periodic?

3.2. Invariant almost periodic algebras on R Let us show that this problem is meaningful even for linear mapping of R: α(x) = qx, q ∈ R. Let A0 be the algebra of continuous functions on R, periodic with the period 1. We will construct the smallest almost periodic algebra A, containing A0 and invariant with respect to this α. As we have already noted, these issues are related to the study of operator (3).

Example 1. Let α(x) = πx. Under the action of operator (W a)(x) = a(πx) on the A0 the functions with frequencies π, π 2 , . . . appear. Due to the fact that number π is transcendental, there are no relations between these frequencies, and the group of frequencies of the smallest invariant algebra A+ is a free group with a countable number of generators π, π 2 , . . . , π k , . . .: H(A+ ) = ZN , M(A+ ) = TN .

Therefore in this case the smallest invariant almost periodic algebra A+ (and A) is not quasi-periodic.

Example 2. Let α(x) = 2x. Then W (A0 ) is the algebra of periodic functions with a period 12 . Since W (A0 ) ⊂ A0 , here A0 is α-invariant and A+ = A0 .

But A0 is not a two-sided invariant. Under the action of W −n the algebra of periodic functions with period 2n is obtained. Therefore the smallest two-sided invariant algebra A is generated by periodic functions with periods 2n and it is not quasi-periodic. Here H(A) =

k : k ∈ Z, n ∈ N 2n

⊂ R.

H(A) is a group with a countable number of generators, but it is not free.

For example, the relations 2n hn = h0 = 1 hold for the “natural” generators hn = 2−n , n = 0, 1, . . . . is called solenoid.

The dual group H(A) The solenoid appeared in many areas. The first it has been found by L. Vietorris in 1927 as an example for the cohomology theory. Van Dantzig (1930) analyzed solenoid as an example of a compact Abelian group with a non-trivial topological

8

A. Antonevich and A. Glaz

structure. It arises as an example of the strange attractor for the system of differential equations (V.V. Nemytskii, V.V. Stepanov, 1940). The role of solinoid in the theory of dynamical systems was detected by S. Smale and R.F. Wilson.

Solenoid can be constructed like the Möbius strip. Let K be the Cantor discontinuum. Solenoid as a topological space can be obtained from the product of [0, 1] × K by identifying {0} × K and {1} × K by means of an invertible map φ : K → K : (0, ω) ∼ (1, φ(ω)). √ Example 3. Let q = 3 + 2 2. Then W (A0 ) is an algebra with a period q. It is easy to check that algebra A with the frequencies group √ √ H(A) = {n + k(3 + 2 2) : n, k ∈ Z} = {n + k2 2 : n, k ∈ Z} is invariant with √ respect to the corresponding α and it is invariant with respect to α−1 (x) = [3 − 2 2]x. We get here the first example of a quasi-periodic algebra, two-sided invariant under a linear mapping.

Note that if we consider a very similar quasi-periodic algebra with a group of frequencies √ {n + k 2 : n, k ∈ Z}, then there is no linear map with respect to which the algebra is two-sided invariant, in other words, there are no non-trivial symmetries.

3.3. Physical automorphisms on Rm The following definitions are similar to the well-known definitions from the number theory. The matrix Q ∈ Cm×m is called algebraic if there is a polynomial P (t) = pn tn + pn−1 tn−1 + · · · + p0 , pk ∈ Z, such that P (Q) = 0. It is called the integer algebraic if pn = 1. Integer algebraic matrix Q is called the algebraic unit if the inverse Q−1 is also an algebraic integer (which is equivalent to pn = 1 and p0 = ±1).

The different structures of the smallest invariant almost periodic algebras from the examined above examples are determined by different algebraic properties of the corresponding numbers q. Indeed, number π is not √ algebraic, numbers 2 is algebraic integer but not an algebraic unit, and q = 3 + 2 2 is an algebraic unit, since it is a root of the polynomial t2 − 6t + 1 = 0.

The next theorem asserts that for arbitrary m the results are similar.

Algebra A0 is called irreducible with respect to Q if minimal vector subspace S of Rm , containing H(A0 ) and invariant with respect to the conjugate map x → QT x is the Rm .

Theorem 3 ([8]). Let A0 be a quasi-periodic algebra on (Rm ), α(x) = Qx and A0 is irreducible with respect to Q. The smallest closed two-sided invariant subalgebra A, that includes A0 , is quasi-periodic if and only if Q is an algebraic unit.

In this case M(A) = TN , and the induced homeomorphism α : TN → TN is an algebraic automorphism of the torus: the covering mapping of RN is given by a

Quasi-periodic Algebras

9

matrix MQ ∈ ZN ×N with determinant ±1. The algebra A is realized as restriction of CP (RN ) on an m-dimensional subspace L invariant with respect to MQ .

Theorem 4. For a given invertible mapping α : Rm → Rm there exists a two-sided α-invariant quasi-periodic algebra A if and only if α can be represented in the form α(x) = Qx + ϕ(x), where Q is an algebraic unit and the mapping ϕ(x) is quasi-periodic (all components are quasi-periodic functions).

References [1] I.A. Dynnikov, S.P. Novikov, Topology of quasi-periodic functions on plane. Uspekhi matem. nauk 60:1(361) (2005), 3–28.

[2] Le Ty Quok Thang, S.A. Piunikhin S.A., Sadov V.A., Geometry of quasi-crystals.

Uspekhi matem. nauk, 1993, 48:1(289), 41–102.

[3] A.B. Antonevich, Linear functional equations. Operator approach. Birkhäuser, 1996.

[4] R. Exel, A new look at the crossed product of a C ∗ -algebra by an endomorphism, Ergodic Theory Dynam. Systems 23(2003) 1733–1750.

[5] A.B. Antonevich, V.I. Bakhtin, A.V. Lebedev, Crossed product by an endomorphism, algebras of coefficients and transfer-operators, Math. Sbornik, 202:9(2011), 3–34.

[6] B.M. Levitan, V.V. Zhikov, Almost periodic functions and differential equations, MGU, Moscow, 1978.

[7] L.S. Pontryagin, Continuous Groups, “Nauka”, Moscow,1973. English transl. Topological Groups, Gordon and Breach, New York, 1066.

[8] A.B. Antonevich, A.N. Glaz, Quasi-periodic algebras invariant with respect to linear mapping, Doklady NAN Belarusi. – 2014. – N. 5. – N. 30–35.

A. Antonevich Institute of Mathematics University of Bia lystok Cio lkowskego 1M 15-245 Bia lystok, Poland e-mail: antonevich@bsu.by A. Glaz Belorusian State University Nezavisimosti 4 220030 Minsk, Belarus e-mail: anna-glaz@yandex.ru

Geometric Methods in Physics. XXXV Workshop 2016 Trends in Mathematics, 11–17 c 2018 Springer International Publishing

Berezin Symbols on Lie Groups Ingrid Beltiţă, Daniel Beltiţă and Benjamin Cahen Abstract. In this paper we present a general framework for Berezin covariant symbols, and we discuss a few basic properties of the corresponding symbol map, with emphasis on its injectivity in connection with some problems in representation theory of nilpotent Lie groups.

Mathematics Subject Classification (2010). Primary 22E27; Secondary 22E25, 47L15.

Keywords. Coherent states, Berezin calculus, coadjoint orbit.

1. Introduction Let V be a finite-dimensional complex Hilbert space and N be a second countable smooth manifold with a fixed Radon measure μ. We denote by L2 (N, V; μ) the complex Hilbert space of (equivalence classes of) V-valued functions μ-measurable on N that are absolutely square integrable with respect to μ. We also endow the space of smooth functions C ∞ (N, V) with the Fréchet topology of uniform convergence on compact sets together with their derivatives of arbitrarily high degree.

If H ⊆ L2 (N, V) is a closed linear subspace with H ⊆ C ∞ (N, V), then the inclusion map H → C ∞ (N, V) is continuous, hence for every x ∈ N the evaluation map Kx : H → V, f → f (x), is continuous. The map K : N × N → B(V),

K(x, y) := Kx Ky∗

is called the reproducing kernel of the Hilbert space H. Then for every linear operator A ∈ B(H) we define its full symbol as K A : N × N → B(V),

K A (x, y) := Kx AKy∗ : V → V

The research of the first two named authors has been partially supported by grant of the Romanian National Authority for Scientific Research and Innovation, CNCS–UEFISCDI, project number PN-II-RU-TE-2014-4-0370.

12

I. Beltiţă, D. Beltiţă and B. Cahen

and K A ∈ C ∞ (N × N, B(V)). See [12, §I.2] for a detailed discussion of this construction, which goes back to [6] and [7].

Main problem In the above setting, the full symbol map B(H) → C ∞ (N × N, B(V)),

A → KA

is injective, as easily checked (see also Proposition 1(1) below). Therefore it is interesting to find sufficient conditions on a continuous map ι : Γ → N × N , ensuring that the corresponding ι-restricted symbol map S ι : B(H) → C(Γ, B(V)),

A → KA ◦ ι

is still injective. The case of the diagonal embedding ι : Γ = N → N × N , x → (x, x), is particularly important and in this case the ι-restricted symbol map is called the (non-normalized) Berezin covariant symbol map and is denoted simply

by S, hence S : B(H) → C ∞ (N, B(V)),

(S(A))(x) := Kx AKx∗ : V → V.

In the present paper we will discuss the above problem and we will briefly sketch an approach to that problem based on results from our forthcoming paper [4]. This approach blends some techniques of reproducing kernels and some basic ideas of linear partial differential equations, in order to address a problem motivated by representation theory of Lie groups (see [8–11]). This problem is also related to some representations of infinite-dimensional Lie groups that occur in the study of magnetic fields (see [1] and [3]). Let us also mention that linear differential operators associated to reproducing kernels have been earlier used in the literature (see, for instance, [5]).

2. Basic properties of the Berezin covariant symbol map In the following we denote by Sp (•) the Schatten ideals of compact operators on Hilbert spaces for 1 ≤ p < ∞.

Proposition 1. In the above setting, if A ∈ B(H), then one has: 1. If A ≥ 0, then S(A) ≥ 0, and moreover S(A) = 0 if and only if A = 0.

2. For all f ∈ H and x ∈ N one has

(Af )(x) = K A (x, y)f (y)dμ(y).

N

3. If {ej }j∈J is an orthonormal basis of H, then for all x, y ∈ N one has

Kx ej ⊗ Ky A∗ ej = ej (x) ⊗ (A∗ ej )(y) ∈ B(V), K A (x, y) = j∈J

j∈J

where for any v, w ∈ V we define their corresponding rank-one operator v ⊗ w := ( · | w)v ∈ B(V).

Berezin Transform 4. If A ∈ S2 (H), then

13

K A (x, y) 2S2 (V) dμ(x)dμ(y)

A 2S2 (H) = N ×N

and if A ∈ S1 (H), then

Tr K A (x, x)dμ(x).

Tr A = N

Proof. See [4] for more general versions of these assertions, in which in particular the Hilbert space V is infinite-dimensional. Assertion (1) is a generalization of [12, Ex. I.2.3(c)], Assertion (1) is a generalization of [12, Prop. I.1.8(b)], while Assertion (1) is a generalization of [12, Cor. A.I.12].

3. Examples of Berezin symbols and specific applications Here we specialize to the following setting: 1. G is a connected, simply connected, nilpotent Lie group with its Lie algebra g, whose center is denoted by z, and g∗ is the linear dual space of g, with the corresponding duality pairing ·, · : g∗ × g → R. 2. π : G → B(H) be a unitary irreducible representation associated with the coadjoint orbit O ⊆ g∗ .

The group G will be identified with g via the exponential map, so that G = (g, ·G ), where ·G is the Baker–Campbell–Hausdorff multiplication.

We use the notation H∞ = H∞ (π) for the nuclear Fréchet space of smooth vectors of π. Let then H−∞ be the space of antilinear continuous functionals on H∞ , B(H∞ , H−∞ ) be the space of continuous linear operators between the above space (these operators are thought of as possibly unbounded linear operators in H), and S(•) and S (•) for the spaces of Schwartz functions and tempered distributions, respectively. Then we have that H∞ → H → H−∞ .

Let X1 , . . . , Xm be a Jordan–Hölder basis in g and e ⊆ {1, . . . , m} be the set of jump indices of the coadjoint orbit O. Select ξ0 ∈ O and let g = gξ0 ge be its corresponding direct sum decomposition, where ge is the linear span of {Xj | j ∈ e} and gξ0 := {x ∈ g | [x, g] ⊆ Ker ξ0 }.

We need the notation for the Fourier transform. For a ∈ S(O) we set

a(x) = e−i ξ,x a(ξ)dξ, O

where on O we consider the Liouville measure normalized such that the Fourier transform is unitary when extended to L2 (O) → L2 (ge ). We denote by F̌ the inverse Fourier transform of F ∈ L2 (g0 ).

14

I. Beltiţă, D. Beltiţă and B. Cahen

Definition 2. 1. For f ∈ H and φ ∈ H, or f ∈ H−∞ and φ ∈ H∞ , let A ∈ C(ge ) ∩ S (ge ) be the coefficient mapping for π, defined by Aφ f (x) = A(f, φ)(x) := (f | π(x)φ), x ∈ ge .

2. For f ∈ H and φ ∈ H, or f ∈ H−∞ and φ ∈ H∞ , the cross-Wigner distribution W(f, φ) ∈ S (O) is defined by the formula φ) = Aφ f.

W(f, Proposition 3. For f, φ ∈ H we have that A(f, φ) ∈ L2 (g0 ), W(f, φ) ∈ L2 (O).

Moreover (A(f1 , φ1 ) | A(f2 , φ2 ))L2 (g0 ) = (f1 | f2 )(φ1 | φ2 ) (W(f1 , φ1 ) | W(f2 , φ2 ))L2 (O) = (f1 | f2 )(φ1 | φ2 ) for all f1 , f2 , φ1 , φ2 ∈ H.

Proof. This follows from [2, Prop. 2.8(i)].

From now on we assume that φ ∈ H∞

with

φ = 1

is fixed.

We let V : H → L (ge ) be the isometry defined by 2

(V f )(x) := (f | φx ) for all x ∈ ge , where φx := π(x)φ. We denote K := Ran V ⊂ L2 (g0 ).

Then K is a reproducing kernel Hilbert space of smooth functions, with inner product equal to the L2 (g0 )-inner product, so the present construction is a special instance of the general framework of Section 1 with V = C. The reproducing kernel of K is given by K(x, y) = (π(x)φ | π(y)φ) = (φx | φy ), and Ky (·) := K(·, y) ∈ Ran V , for all y ∈ g0 . We also note that (∀x ∈ g0 ) Kx = V φx .

The Berezin covariant symbol of an operator T ∈ B(K) is then the bounded continuous function S(T ) : ge → C,

S(T )(x) = (T Kx | Kx )K .

One thus obtains a well-defined bounded linear operator S : B(K) → C ∞ (ge ) ∩ L∞ (ge ) which also gives by restriction a bounded linear operator S : S2 (K) → L2 (g0 ).

Berezin Transform

15

To find accurate descriptions of the kernels of the above operators is a very important problem for many reasons, as explained in [8–11] also for other classes of Lie groups than the nilpotent ones.

The case of flat coadjoint orbits of nilpotent Lie groups We now assume that the coadjoint orbit O is flat, hence its corresponding representation π is square integrable modulo the center of G. Remark 4. Consider the representation ρ : G → B(K), ρ(g) = V π(g)V ∗ , that is a unitary representation of G equivalent to π, thus it corresponds to the same coadjoint orbit O. We denote by Opρ the Weyl calculus corresponding to this representation. The following then holds: 1. For a ∈ S (O) one has Opρ (a) = V Op(a)V ∗ = Ta .

2. For T ∈ B(K) and X ∈ g0 , one has S(ρ(x)−1 T ρ(x))(z) = S(T )(x · z),

for all z ∈ g0 .

(1)

Theorem 5. Assume that in the constructions above, φ ∈ H∞

is such that W(φ, φ) is a cyclic vector for α.

(2)

Then S : S2 (K) → L2 (g0 ) is injective.

Proof. The method of proof is based on specific properties of the Weyl–Pedersen calculus from [2].

We refer to [4] for a more complete discussion and for proofs of the above assertions in a much more general setting. To conclude this paper we will just briefly discuss an important example.

The special case of the Heisenberg groups Let G be the Heisenberg group of dimension 2n + 1 and H be the center of G. Let {X1 , . . . , Xn , Y1 , . . . , Yn , Z} be a basis of g in which the only nontrivial brackets are [Xk , Yk ] = Z, 1 ≤ k ≤ n and let {X1∗ , . . . , Xn∗ , Y1∗ , . . . , Yn∗ , Z ∗ } be the corresponding dual basis of g∗ .

For a = (a1 , a2 , . . . , an ) ∈ Rn , b = (b1 , b2 , . . . , bn ) ∈ Rn and c ∈ R, we n n denote by [a, b, c] the element expG ( k=1 ak Xk + k=1 bk Yk + cZ) of G. Then the multiplication of G is given by 1 [a, b, c][a , b , c ] = [a + a , b + b , c + c + (ab − a b)] 2 and H consists of all elements of the form [0, 0, c] with c ∈ R.

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Ernst Stern, his notable designer; there are the horrors of Poelzig’s decoration of the Grosses Schauspielhaus to testify to this.

The Cathedral Scene from Faust. A Reinhardt production of 1912, designed by Ernst Stern. Two huge columns tower up against black emptiness. Crimson light from the unseen altar at one side streams on the congregation and throws quivering shadows of a cross on the nearer column.

The three moments of Orpheus which electrified Swedish audiences are common enough in conception, but they have something of the simple directness and smash which characterized Reinhardt’s earlier work. The three episodes are closely linked and

make the climax of the piece. There again you can see Reinhardt’s method—the expenditure of so much of his care and energy upon the most important action of the play. In Orpheus the place for such emphasis is the revolt on Mt. Olympus, and the descent of Jupiter and the gods to Hades. Reinhardt begins with the carmagnole of the revolutionists, with their red banners upon long poles rioting about in the light blue of the celestial regions. For the beginning of the descent into Hades, Reinhardt sees to it that there shall be a high point at the very back of the stage, and from here, clear down to the footlights and over them on a runway beside the boxes, he sends his gods and goddesses cakewalking two at a time down into the depths of the orchestra pit. After a very brief darkness, while the cloud and its rays of light are installed down stage, Reinhardt sets the gods prancing down this white and black path into the flaming silk mouth of hell. By recognizing an opportunity for an effect at the crucial point of the piece, and concentrating upon it whatever energies he has for Orpheus, he makes the descent of the gods far more memorable than it can have been in any other production. Yet it all seems a trivial and half-hearted effort for the man who made Shakespeare so tremendously vital at the Deutsches Theater, and lifted Sophocles’ Œdipus into crashing popularity at the Circus Schumann.

In his day Reinhardt was all things to all men. He began with the great naturalist director Brahm of the Freie Volksbühne. He made a Night Lodging of utter Realism. He put on A Midsummer Night’s Dream in a forest of papier-mâché. He brought an austere symbolic quality to Hamlet, closing the play with those tall, tall spears that shepherded the body of the Dane upon its shield. He made the story of Sister Beatrice into a gigantic and glorious spectacle in The Miracle. He championed intimacy in the theater, took the actor out upon a runway over the heads of the audience in Sumurûn and finally, at the Grosses Schauspielhaus, he put the spectators half around the players, and thrust the players in among the spectators in the last scene of Rolland’s Danton.

Instinct led him to the heart of plays, as it led him from Realism and the proscenium frame back to the Greek orchestra and the actor as a theatrical figure. He grasped the emotional heart of a drama

with almost unerring judgment, and he bent a tremendous energy to the task of making the heart of the audience beat with it.

Occasionally he ignored or could not animate some secondary but important phase of a play. In The Merchant of Venice, though he made Shylock rightly the center of the play and built up a court scene of intolerable excitement, his Portia and his Nerissa were tawdry figures. But his successes were far greater and far more significant than his failures. Romeo and Juliet he made into a thing of youthful passion that was almost too deep, too intimate for the eyes of strangers. Hamlet with Moissi was an experience of life itself, asserting again the emotional quality of Reinhardt as against the esthetic quality of Craig.

It is hardly necessary to speak of the part that Reinhardt played in establishing the vogue of the designer in the theater, of his attempt to bring Craig to his stage, of his experiments with stage machinery and lighting equipment, or of the extraordinary personal energy which made so much work possible. The German theater testifies continually to his influence. Dozens of younger men must be working in his vein to-day. As far north as Gothenburg, the commercial city of Sweden, and as far south as Vienna his influence spreads.

In Gothenburg works a young director, Per Lindberg, who is as patently a disciple as he was once a student of Reinhardt. There in the Lorensberg Theater is the revolving stage, with settings by a young Swede, Knut Ström, which might have been seen at the Deutsches Theater ten years ago. A large repertory brings forth scenery often in the heavily simplified fashion of ten years ago, but sometimes fresh and ambitious. Romeo and Juliet appears against scenes like early Italian paintings, with one permanent background of hill and cypresses and a number of naïve arrangements of arched arcades from some Fra Angelico. The artist turns régisseur also in Everyman, and manages a performance fresh in its arrangement of setting, platforms, and steps, if a little reminiscent in costumes and poses and movements.

In Richard Weichert, of the State Schauspielhaus in Frankfort, you find a régisseur who suggests the influence of Reinhardt without losing distinction as one of the three really significant directors of

Germany to-day. It is not so much an influence in an imitative sense, as a resemblance in effectiveness along rather similar lines.

Maria Stuart: the throne-room at Westminster. Tall screens of blue and gold are ranged behind a dais surmounted by a high, pointed throne of dull gold. At either side curtains of silvery blue. Queen Elizabeth wears a gown of gleaming gold. A Weichert production in Frankfort designed by Sievert.

Weichert, like so many of the outstanding directors of Germany, has a single artist with whom he works on terms of the closest coöperation—Ludwig Sievert. It is a little hard, therefore, to divide the credit in Maria Stuart for many of the dramatic effects of people against settings and in light. You might put down the scenic ideas wholly to Sievert, since Weichert has permitted the use of a particularly poor setting for the scene of Queen Mary’s tirade against

Elizabeth; a setting which is a sloppy attempt at lyricism in keeping with Mary’s speech at the beginning of the scene, but quite out of touch with the dramatic end. If Weichert could dictate the fine prison scene reproduced in this book, he would hardly allow Sievert to include the greenery-yallery exterior to which I have taken exception.

On the other hand, can it be only an accidental use that Weichert makes of the curtains in the throne room scene? The act begins with a curious arrangement of square blue columns in an angle of which the throne is set. When the audience is over, pages draw blue curtains from each side of the proscenium diagonally backward to the columns by the throne. This cuts down the room to terms of intimacy for the council scene. The point at which Weichert must enter definitely as régisseur comes when Elizabeth steps to one side of the room away from her group of councilors to read some document; then the down-stage edge of the curtain at the side by the councilors is drawn back far enough for a flood of amber light to strike across in front of the men, and catch the white figure of the queen. Here in this light she dominates the room; and Leicester, when he steps into it for a scene with Mortimer, does the same. It is a device of great use to the actor in building up the power and atmosphere of the moment.

The dramatic vigor of Weichert never goes so high in Maria Stuart as Reinhardt’s, but he is never so careless of detail or of subordinate scenes. Almost every inch of the play seems painstakingly perfected.

Not only are the actors who give so sloppy a performance in Peer Gynt under another director, strung up constantly to their best effort; but every detail, from contrasts in costuming and the arrangement of costumed figures, to the motion of hands and bodies, seems calculated to heighten the play’s emotion. Take the first scene, for example, the prison in which Queen Mary is confined with her few retainers. The drawing shows the interesting arrangement of the scene with bars to indicate a prison but not to obstruct action. It pictures the final scene in a later act, when the queen receives her friends and says good-by before going to her death. The contrast of the queen in white and the others in black is excellent. In the first act, even the queen is in black; the only note of color, a deep red, is given to the heroic boy, Mortimer, who is to bring something like

hope to Mary. The long scene between Mortimer and the queen is handled with great dignity, and at the same time intensity. It is studied out to the last details. The hands alone are worth all your attention.

Weichert’s direction passes on from atmosphere and movement to the expression that the players themselves give of their characters. It is here perhaps that the resemblance to Reinhardt is closest. You catch it in many places: the contrast between Mortimer’s tense young fervor, and the masterful, play-acting nonchalance of Leicester; this red and green horror of an Elizabeth, looking somehow as bald beneath her wig as history says she was, and bursting with pent energies and passions; towards the end of the play, Leicester, the deliberate fop, leaning against the wall like some wilted violet, Mortimer exhausted but still strong beside him; then the death of the boy, the quick stabbing, and the spears of the soldiers raying towards his body on the floor. It is all sharp, firm, poised—and very, very careful.

Maria Stuart: a room in the castle where Queen Mary is imprisoned.

High black grills fill the proscenium arch on either side. Behind, a flat wall of silvery gray. The sketch shows the moment when Mary, gowned and veiled in white, bids farewell to her attendants. A Weichert production in Frankfort designed by Sievert.

This is the past of Reinhardt—continued into the present and the future by other men. What of his own continuation of it? Some have thought him finished. Fifteen, twenty years of such accomplishment in the theater are likely to drain any man. And indeed Reinhardt does seem to have run through his work in Berlin, and finished with it. No one will know just how much was personal, how much professional, how much philosophic, in the force that drove him to give up the leadership of his great organization, and see it destroyed. The difficulties of management, with increasing costs and actors lost to the movies, undoubtedly weighed heavily. But it is certain that he felt the failure of his big, pet venture, the Grosses Schauspielhaus. It

was to have been the crown of his efforts and beliefs—the “theater of the five thousand,” as he had called it from the days when he astounded the world with Œdipus. In structure and design it was badly handled; it proved a bastard thing and won the severe condemnation of the critics. Added to this was a desire, unquestionably, to shake loose, to get a fresh prospect on the theater, to strike out again if possible towards a final, sure goal.

Germans spoke of Reinhardt as vacillating and uncertain in his first years in Salzburg. But is anything but uncertainty to be expected when a man has given up a long line of effort, and is seeking a new one? It is a virtue then to be unsure, to be testing and trying the mind, to be seeking some sort of truth and repeatedly rejecting error.

Certainty began to creep in with Reinhardt’s plan for a Festspielhaus in Salzburg—a Grosses Schauspielhaus of simpler and more conservative pattern built truer on a knowledge of the mistakes of the first. It was to unite Reinhardt, Richard Strauss, the composer, and Hugo von Hofmannsthal, the playwright. It reached some sort of tentative plan at the hands of Poelzig, who misdesigned the Grosses Schauspielhaus, and Adolf Linnebach, then passed on to Max Hasait, who laid out a stage scheme for some new architect to build his plans around. This scheme called for a semicircular forestage, with a revolving stage in its center, a traveling cyclorama of the Ars pattern behind this revolving stage, a larger cyclorama taking in still a deeper stage, and another and a larger cyclorama behind that. The proscenium was to be narrowed or widened to suit the size of production and cyclorama. The house itself was to be as adjustable, with a ceiling that let down in such a way as to cut the seating capacity from three or four thousand to fifteen hundred.

While this project waited on capital, an almost hopeless condition in Austria, and hints began to come that the Festspielhaus would have to be built in Geneva instead, a new opportunity came to Reinhardt’s hands through President Vetter, head of the Austrian State theaters, an opportunity of working in a playhouse that agreed with much that Reinhardt had felt about the relations of audience and actor. He was invited to produce five or six plays in the fall of 1922 in

the new theater in the Redoutensaal in Vienna. Here, upon a stage practically without setting, and within a room that holds actors and audience in a matrix of baroque richness, Reinhardt will have produced, by the time this book appears, the following plays: Turandot, Gozzi’s Italian comedy, Clavigor and Stella by Goethe, Molière’s Le Misanthrope, and Dame Cobalt by Calderon. Here he will have to work in an absolutely non-realistic vein, he will have to explore to the fullest the possibilities of the new and curious sort of acting which I have called presentational. This adventure in Maria Theresa’s ballroom will measure Reinhardt against the future.

CHAPTER X THE ARTIST AS DIRECTOR

T

HE director of the future may not be a director of to-day. He may not be a director at all. He may be one of those artists whose appearance has been such a distinctive and interesting phenomenon of the twentieth century theater. While we examine Max Reinhardt to discover if he is likely to be the flux which will fuse the expressionist play and the presentational actor, it may be that the man we seek is his former designer of settings, Ernst Stern.

The relation of artist and director in the modern theater has been a curious one, quite as intimate as that of pilot-fish and shark, and not so dissimilar. Attached to the shark, the pilot-fish has his way through life made easy and secure; he is carried comfortably from one hunting ground to another. Often, however, when the time comes to find food, it is the pilot-fish that seeks out the provender, and prepares the ground, as it were, for the attack of the shark. Then they both feast, and the pilot-fish resumes his subordinate position.

We may shift the figure to pleasanter ground by grace of Samuel Butler, the Erewhonian. This brilliant, odd old gentleman, a bit of a scientist as well as a literary man, had a passion for transferring the terms and conceptions of biology to machinery and to man’s social relationships. Departing from the crustaceans, which grow new legs or tails as fast as the old are cut off, he said: “What ... can be more distinct from a man than his banker or his solicitor? Yet these are commonly so much parts of him that he can no more cut them off and grow new ones than he can grow new legs or arms; neither must he wound his solicitor; a wound in the solicitor is a very serious thing. As for his bank,—failure of his bank’s action may be as fatal to a man as failure of his heart.... We can, indeed, grow butchers, bakers, and greengrocers, almost ad libitum, but these are low developments and correspond to skin, hair, or finger nails.”

I do not know whether it would be right to say that directors have grown artists with great assiduity in the past twenty years, or that the greatest of the directors have become as closely associated with particular artists as a well-to-do Englishman is with his banker or his solicitor. At any rate the name of Reinhardt is intimately associated with the name of Stern; Jessner has his Pirchan, Fehling his Strohbach; I have spoken of the close relationship of Weichert and Sievert, and I could point out similar identifications in America. An artist of a certain type has come into a very definite, creative connection with the art of production, and he has usually brought his contribution to the theater of a particular director.

The designer is a modern product. He was unknown to Molière or Shakespeare; the tailor was their only artist. Except for incidental music, costume is the one field in which another talent than that of actor or director invaded the theater from Greek days until the last years of the seventeenth century. There were designers of scenery in the Renaissance, but they kept to the court masques. Inigo Jones would have been as astonished and as shocked as Shakespeare if anybody had suggested that he try to work upon the stage of the Globe Theater. The advent of Italian opera—a development easy to trace from the court masques—and the building of elaborate theaters to house its scenery, brought the painter upon the stage. The names of the flamboyant brothers Galli-Bibiena are the first great names to be met with in the annals of scene painting. And they were the last great names until Schinkel, the German architect, began in the early nineteenth century to seek a way of ridding the stage of the dull devices of the current scene painters. Scenery was not an invention of Realism; it was a much older thing. I doubt if any one more talented than a good carpenter or an interior decorator was needed to achieve the actuality which the realists demanded. When artists of distinction, or designers with a flair for the theater appeared at the stage door, it was because they saw Shakespeare or Goethe, von Hofmannsthal or Maeterlinck sending in their cards to Irving or Reinhardt or Stanislavsky.

The Desert: a setting by Isaac Grünewald from the opera, Samson and Delilah. A vista of hills and sky, painted and lit in tones of burning orange, is broken at either side by high, leaning walls of harsh gray rock. The director, Harald André, has grouped his players so as to continue the triangular form of the opening through which they are seen. At the Royal Opera in Stockholm.

Now what are the relations that this modern phenomenon has established with the theater through the medium of the director?

Ordinarily they differ very much from the attitude that existed between the old-fashioned scenic artist and the director, and the attitude that still exists in the case of most scenic studios. This is the relation of shopkeeper and buyer. The director orders so many settings from the studio. Perhaps he specifies that they are to be arranged in this or that fashion, though usually, if the director hasn’t

the intelligence to employ a thoroughly creative designer, he hasn’t the interest to care what the setting is like so long as it has enough doors and windows to satisfy the dramatist. Occasionally you find a keen, modern director who, for one reason or another, has to employ an artist of inferior quality. Then it is the director’s ideas and conceptions and even his rough sketches and plans that are executed, not the artist’s. In Stockholm, for example, Harald André so dominates the official scene painter of the Opera that the settings for Macbeth are largely André’s in design though they are Thorolf Jansson’s in execution. Even in the case of the exceptionally talented artist, Isaac Grünewald, with whom André associated himself for the production of Samson and Delilah, the director’s ideas could dominate in certain scenes. For example, in the beautiful and effective episode of the Jews in the desert which André injected into the first act—a scene for which the director required a symbolic picture of the fall of the walls of Philistia to accompany the orchestral music which he used for this interlude. The brilliance with which Grünewald executed the conception may be judged from the accompanying illustration.

The commonest relationship of the director and the designer has been coöperative. The artist has brought a scheme of production to the director as often, perhaps, as the director has brought such a scheme to the artist. The director has then criticized, revised, even amplified the artist’s designs, and has brought them to realization on the stage. And the artist and the director, arranging lights at the final rehearsals, have come to a last coöperation which may be more important to the play than any that has gone before.

Samson and Delilah: the mill. A remarkable example of an essentially ornamental theatrical setting, designed by Isaac Grünewald for the Royal Opera in Stockholm. Black emptiness. A slanting shaft of light strikes the millstone in a vivid crescent. As the wheel travels in its track this crescent widens to a disk of blinding light, and then shrinks again. The actual forms of this setting are sublimated into an arresting composition of shifting abstract shapes of light.

You find, however, constant evidence of the artist running ahead of the director in the creation of details of production which have a large bearing on the action as well as on the atmosphere of the play.

Grünewald brought a setting to the mill scene in Samson and Delilah which was not only strikingly original and dramatic, but which forced the direction into a single course. The usual arrangement is the flat

millstone with a long pole, against which Samson pushes, treading out a large circle as the stone revolves. The actor is always more or less visible, and there is no particular impression of a cruel machine dominating a human being. Grünewald changed all this by using a primitive type of vertical millstone. The sketch shows the stage in darkness except for one shaft of light striking sideways across. The great wheel is set well down front within a low circular wall. Along the wall Samson walks, pushing against a short pole that sticks out from the center of one face of the high narrow, millstone. As he pushes, the stone swings about and also revolves. This allows the beam of light to catch first a thin crescent at the top of the curving edge of the wheel, then a wider and wider curve, until suddenly, as Samson comes into view, the light brings out the flat face of the wheel like a full moon. Against this the actor is outlined for his aria. Then, while the orchestra plays, he pushes the wheel once more around. This arrangement is extraordinarily fine as a living picture and as an expression of the mood of the scene. Moreover, it is a triumph for the artist, because it is an idea in direction as well as setting. It dictates the movement of the player and manages it in the best possible way.

No other action for Samson is possible in this set, and no other action could be so appropriate and effective.

Examples of similar dictation by the artist—though none so striking —come to mind. In Frankfort Sievert arranged the settings for Strindberg’s Towards Damascus in a way that contributed dramatic significance to the movement of the players. The piece is in seventeen scenes; it proceeds through eight different settings to reach the ninth, a church, and from the ninth the hero passes back through the eight in reverse order until he arrives at the spot where the action began. Sievert saw an opportunity to use the revolving stage, as well as elements of design, in a way interpreting and unifying the play. He placed all nine scenes on the “revolver,” and he made the acting floor of each successive setting a little higher than the last. This results in rather narrow rooms and a sea shore bounded by formal yellow walls, but it permits an obvious unity, it shows visually the path that the hero has to follow, and it symbolizes his progress as a steady upward movement towards the church.

The artist dictating a particular kind of direction is obvious enough in Chout (Le Bouffon), the fantastic comic ballet by Prokofieff which Gontcharova designed for the Ballets Russes. Gontcharova’s settings are not particularly good, but at least they have a definite and individual character. They are expressionist after a fashion related more or less to Cubism. They present Russian scenes in wildly distorted perspective. Log houses and wooden fences shatter the backdrop in a war of serried timbers. A table is painted on a wing, the top tipping up at an alarming angle, one plate drawn securely upon it, and another, of papier-mâché, pinned to it. All this sort of thing enjoined upon the régisseur a kind of direction quite as bizarre, mannered, and comic. Chout seems to have had no direction at all in any creative sense. The régisseur failed to meet the challenge of the artist.

The first scene of Tchehoff’s Uncle Vanya. Here Pitoëff indicates a Russian country side by a rustic bench and slender birch trees formally spaced against a flat gray curtain.

It is ordinarily very hard to say what share the artist or the director has had in the scheme of a setting, or whether the director has bothered his head at all about the setting after confiding it to what he considers competent hands. It is an interesting speculation just how much the physical shape of Reinhardt’s productions has been the sole creation of his artist, Stern. Certainly Stern delighted in the problems which the use of the revolving stage presented, and only in a single mind could the complexities of these sets, nesting together like some cut-out puzzle, be organized to a definite end. It is entirely possible that, except for a conference on the general tone of the

production, and criticisms of the scheme devised by Stern, Reinhardt may have given no thought at all to the scenery. Stern was a master in his own line, and for Reinhardt there was always the thing he delighted most in, the emotional mood produced by the voices and movements of the actors. His carelessness of detail even in the acting, suggests that for him there were only the biggest moments, the important elements and climaxes, that put over the emotion of the play.

Sometimes artist and director are the same, as with Pitoëff in Geneva and Paris, or with Knut Ström in Gothenburg. In such a case setting, direction, and acting are one. But ordinarily there is a division of responsibility, and an opportunity for the artist to play a part in the production of a drama far more important than Bibiena’s. Just how important it may prove to be is bound up, I think, with the future of the theater as a physical thing, and with the temperament of the artist. Working as a designer of picture-settings, the artist can only suggest action, but not dictate it, through the shapes and atmospheres he creates. The important thing is that almost all the designers of real distinction in Europe are tending steadily away from the picture-setting. They are constantly at work upon plans for breaking down the proscenium-frame type of production, and for reaching a simple platform stage or podium upon which the actor shall present himself frankly as an actor. This means, curiously enough, that the designers of scenery are trying to eliminate scenery, to abolish their vocation. And this in turn should indicate that the artist has his eye on something else besides being an artist.

The director who works in such a new theater as the artists desire —in the Redoutensaal in Vienna, for example,—requires an artist to work with him who sees art in terms of the arrangement of action upon steps, and against properties or screens. This is ordinarily the business of the director in our picture-frame theater; with the work of the artist enchantingly visible in the setting behind the actors, the director can get away reasonably well with the esthetic problems of the relations of actors and furniture and of actors and actors. Nobody notes his shortcomings in this regard. Put him upon an almost naked stage, and he must not only make his actors far more expressive in

voice and feature, but he must also do fine things with their bodies and their meager surroundings. This is far easier for a pictorial artist than for the director, who is usually an actor without a well-trained eye. The director must therefore employ an artist even in the sceneryless theater, and employ him to do what is really a work of direction. The two must try to fuse their individualities and abilities, and bring out a composite director-artist, a double man possessing the talents that appear together in Pitoëff.

A scene from Grabbe’s Napoleon. The Place de Grêve in Paris is indicated by a great street lamp set boldly on a raised platform in the center of the stage. A Jessner production designed by Cesar Klein.

The immediate question is obviously this: If the director cannot acquire the talents of the artist, why cannot the artist acquire the talents of the director? If the knack of visual design, and the keen

appreciation of physical relationships cannot be cultivated in a man who does not possess them by birth, is it likewise impossible for the man who possesses them to acquire the faculty of understanding and of drawing forth emotion in the actor?

The problem narrows down to the temperament of the artist versus the temperament of the director. There is a difference; it is no use denying it. The director is ordinarily a man sensitive enough to understand human emotion deeply and to be able to recognize it, summon it, and guide it in actors. But he must also be callous enough to meet the contacts of direction—often very difficult contacts—and to organize not only the performance of the players, but also a great deal of bothersome detail involving men and women who must be managed and cajoled, commanded, and worn down, and generally treated as no artist cares to treat others, or to treat himself in the process of treating others. The director must be an executive, and this implies a cold ability to dominate other human beings, which the artist does not ordinarily have. The artist is essentially a lonely worker. He is not gregarious in his labor.

So far as the future goes, the hope for the artist is that he will be able to reverse the Butlerian process which held in the relations of director and designer. He must be able to “grow a director.” This may not be so very difficult. It may very well happen that an artist will employ a stage manager, as an astute director now employs an artist, to do a part of his work for him. He will explain to the stage manager the general scheme of production that he wants, much as a director explains to an artist the sort of settings he desires. The stage manager will rehearse the movements of the actors towards this end. When the artist sees opportunities for further development of action and business, he will explain this to the stage manager, and perhaps to the players involved, and the stage manager will again see that the ideas of his superior are carried out. Something of the kind occurs even now where a director employs a subdirector to “break in” the company. Both Reinhardt and Arthur Hopkins, though thoroughly capable of “wading into” a group of players, and enforcing action by minute direction and imitation, generally use the quiet method of consulting with players, and suggesting changes to them,