Root structure of a special generalized kac moody algebra

Mathematical Computation September 2014, Volume 3, Issue 3, PP.83-88

Root Structure of a Special Generalized KacMoody Algebra Xinfang Song1, #, Xiaoxi Wang2 1. Basis Department, Beijing Information Technology College, Beijing, 100070, China 2. College of Computer Science and Technology, Beijing University of Technology, Beijing, 100121, China #Email: xf-sohu@sohu.com

Abstract The paper systematically discusses the root structure of a special generalized Kac-Moody algebra EB2 which was extended from the finite Kac-Moody algebra B2 . It’s extended one point from the finite Coxter diagram B2.At the beginning the paper define a special generalized generalized Cartan matrix (abbreviated as GGCM) and a special generalized Kac-Moody algebra(abbreviated as GK algebra). As a starting point for the special GK algebra it mainly obtain the special imaginary root system, the relationship between reflections determined by imaginary roots and the Weyl group for the special GMC algebra and the purely and the strictly imaginary roots.

Keywords: Generalized Kac-Moody Algebra, Imaginary Root, Special Imaginary Root, Weyl Group, Strictly Imaginary Root, Purely Imaginary Root

1 INTRODUCTION This paper continues to use the concept of the special root and the existence of the imaginary roots from [1]. It follows the imaginary roots and the special root for a class of generalized Kac-Moody algebra with rank 3 from [2].

And also it follows the relationship between reflections determined by imaginary roots and the Weyl group for a special GKM algebra. In [2] the author mainly discussed the generalized Kac-Moody algebra from the extension of finite type A2. In the next section, after a statement of the basic problem, various situations involving possibility knowledge are investigated: first, an entirely possibility model is proposed; then the cases of a fuzzy service time with stochastic arrivals and non fuzzy service rule is studied; lastly, fuzzy service rule are considered.This present paper enhances the degree of difficulty being obtained from the extension of finite type GCM B2.

Note: The paper also uses the symbol from [2].

2 BASIC CONCEPTS AND SYMBOL In this part the paper gives some basic definitions (from [1]) regarding Kac-Moody algebra and GKM algebra.

Definition 2.1 Let A  (aij )nn be a real n  n matrix satisfying the following conditions: (c1) either aii  2 or aii  0 ; (c2) aij  0 if i  j and aij  Z if a ii  2 (c3) aij  0  a ji  0 .

A is called a generalized generalized Cartan matrix (abbreviated as GGCM). And the Lie algebra g(A) associated with A is called the generalized Kac-Moody algebra (abbreviated as GKM algebra).

Note 2.1 Here assume that the entries in the GGCM are integers.

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Definition 2.2 For a GGCM A, (  ) be a realization of A, where   {1   {1 

  n }    and

 n }   are independent sets. n

n

i 1

i 1

Note 2.2 Q   Zi and Q   Zi denotes the root lattice and the positive root lattice of g(A) respectably.

Let  (resp.   ) be the root system (resp. positive root set) of g(A). Denote re  {i  | aii  2} to be the set of real simple root and im  {i  | aii  0} the set of simple imaginary root. Denote W   i | i  re

by the

Weyl group of g(A).

Definition 2.3 We define the real (resp.imaginary) root set of g(A)to be re  W ( re )( resp. im   define C   {   | i    0i  } to be the dual fundamental Weyl chamber and N  Z

re ) . We

{0} to be the

set of natural numbers. And denote R  to be the set of real positive numbers.

Let K0  {  Q \ {0}|  C  and supp  is connected} and K  K0  j 2 jim .

In particular the set of positive imaginary roots is W-stable.

In [6] we have the following propositions: Proposition 1: im  

W

(K )

A GGCM A is called symmetralized if there exists an invertible diagonal matrix D  diag (1   n ) with  i  R  for all i and a symmetric matrix B such that A=DB. There exists a non-degenerate, symmetric, invariant bilinear form ( , )on  . We have an isomorphism       defined by  (h) h1  (h h1 ) h h1  and the induced bilinear form ( , ) on   . It’s clear that  (i )   ii , (i  j )  bij  i j  1

 n . We note that a root    is imaginary if

and only if (  )  0 .

Proposition 2: 2   1 (i )   ( i  i )  i   2   1 (i )   ( i  i )

aii  2 aii  2

3 THE MAIN RESULTS 3.1 The Imaginary Root System Of A Class Of Rank=3 GKM Algebra  2 a 2a  Lemma 1: Let A   b 2 2   where a, b  N. Then  b 1 2    K  {1}  {k11  k22 | 2k2  ak1 k1 k2  N} {k11  k33 | k3  ak1 k1 k3  N} {k11  k22  k33 | 2k2  ak1  k3  k3  k2  ak1 k1  N  k2  k3 are not zero at the same time}.

Proof: Since im  {1}re  { 2 3} and C   {   | i    0 i  1 2} Suppose   k11  k22  k33  Q Then

{0}

 2   ak1  2k2  k3  0 thus 2k2  ak1  k3 ,  3   2ak1  2k2  2k3  0 thus k3  k2  ak1 .

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And K0  {  Q {0}|  C  and supp  is connected} 3

 {   kii  Q {0}| 2k2  ak1  k3  k3  ak1  k2 } i 1

3

It’s clear that k1  0 for every    kii  K0 . i 1

If k2  k3  0 , then   k11  K0  k1  N .

So K0  (

jim ) j 1

{k11  k2 2  k33 | 2k2  ak1  k3  k3  ak1  k2  k1  N  k2 and k3 are not zero at the same

time}.

Since K  K0

jim , we get the proof of lemma 1. j 2

For the simplicity, we let p(k1 k2  k3 ) denote the set of all k1 k2 and k3 satisfying the conditions: 2k2  ak1  k3  k3  ak1  k2  k1  N  k2 and k3 are not zero at the same time. Then we can get K in lemma 1 as follows: K  {1} {k11  k22  k33 | p(k1 k2  k3 )}

 2 a 2a  Lemma 2: Let A   b 2 2   where a, b  N. Then the Weyl group of g(A) is  b 1 2    W  {1  2   3   2 3   3 2   2 3 2   3 2 3  ( 2 3 )2 } .

Proof: We know that W   i | i  re   2   3  {1  2   3   2 3   3 2   2 3 2   3 2 3  ( 2 3 )2 } .

The result can be easily obtained in operation by the GGCM A.

 2 a 2a  Theorem 1: Let A   b 2 2   where a, b  N. Then the positive imaginary root set of g (A) is  b 1 2   

im  

W

 ( K )  W (1 ) W (k11  k2 2  k33 ) , where K from lemma 1.

Proof: From Lemma 2 we know the Weyl group of g(A) is W  {1  2   3   2 3   3 2   2 3 2   3 2 3  ( 2 3 )2 } We get:

 2 ( K )  {1  b 2 }  {k11  (bk1  k2  2k3 ) 2  k33 | p(k1 k2  k3 )}  3 ( K )  {1  b 3}  k11  k2 2  (bk1  k2  k3 ) 3 | p(k1 k2  k3 )}  2 3 ( K )  {1  3b 2  b3}  {{k11  (3bk1  k2  2k3 ) 2  (bk1  k2  k3 )3 | p(k1  k2  k3 )}  3 2 ( K )  {1  b 2  2b 3}  {k11  (bk1  k2  2k3 ) 2  (2bk1  k2  k3 )3 p(k1 k2  k3 )}  2 3 2 ( K )  {1  4b 2  2b 3}  {k11  (2bk1  k2  2bk3  k3 ) 2  (bk1  k2  k3  bk3 ) 3 | p(k1 k2  k3 )}  3 2 3 ( K )  {1  3b 2  3b 3}  {k11  (3bk1  k2 ) 2  (2ak1  k2  k3 ) 3 | p(k1 k2  k3 )} ( 2 3 )2 ( K )  {1  4b 2  3b 3}  {k11  (4bk1  3k2  2k3 ) 2  (3bk1  k2 )3 | p(k1 k2  k3 )}.

As

2 im   K   2 ( K )   3 ( K )   2 3 ( K )   3 2 ( K )   2 3 2 ( K )   3 2 3 ( K )  ( 2 3 ) ( K )

 {11  b2 1  b3 1  3b2  b3 1  b2  2b3 1  4b2  2b3 1  3b2  3b3  1  4b2  3b3}

{k11  k22  k33  k11  (bk1  k2  2k3 )2  k33  k11  k22  (bk1  k2  k3 )3 

k11  (3bk1  k2  2k3 )2  (bk1  k2  k3 )3  k11  (bk1  k2  2k3 )2  (2bk1  k2  k3 )3 

k11  (2bk1  k2  2bk3  k3 )2  (bk1  k2  k3  bk3 )3  k11  (3bk1  k2 )2  (2ak1  k2  k3 )3  k11  (4bk1  3k2  2k3 )2  (3bk1  k2 )3 | p(k1 k2  k3 )} - 85 www.ivypub.org/mc

So we obtain the proof of this theorem.

3.2 Purely Imaginary Roots and Strictly Imaginary Roots  2 a 2a  Suppose A   b 2 2  ,where a b  N   b 1 2   

There

is

a

 2b B   ab  ab 

diagonal

ab 2a 2a

matrix D

1   (b )   0 0  

0 0  1 (a ) 0  , 0 (a )1 

with

  R

and

a

symmetric

matrix



2ab  2a  .So A=DB exits. Then GGCM A is symmetrizable.

4a 

Definition 1: An imaginary root  in a GKM algebra is said to be strictly imaginary root if for every    re , either

   or    is a root. The set of all strictly imaginary roots is denoted by  sim . im im Definition 2: Let   im  , define  to be purely imaginary root if for any            . The set of all

purely imaginary roots is denoted by  pim .

Proposition 3: Let A  (aij )ijn 1 be an indecomposable symmetrizable GGCM such that aij  0 i j  N and i  j.Then pim  im  .

Proof: In [4].

 2 Proposition 4: Consider the GKM algebra associated with the GGCM A   b1  b  2

a1 a2  2  p  with pq  3 and q=1. q 2 

pim  sim .

If both b1 and b2  1 . pim  im  .Then  

Proof: In [4].

 2 a 2a  Theorem 2: Let A   b 2 2   where a b  N Then pim  sim  im  .  b 1 2    Proof: It’s easily known that A is an indecomposable symmetrizable GGCM. And every aij  0 i j  N and i  j.

From proposition 3, we get pim ( A)  im  ( A) . And also A is satisfied with proposition 4. Then theorem 2 is completed the proof.

3.3 Reflections Determined By the Imaginary Roots and the Weyl Group for a Special GKM Algebra Let g(A) be a symmetrizable GKM algebra ,  be an imaginary root of g(A) . If (  )  0 then we define a reflection on   by   ( )    (2 )  1 ( ) for    . If we set    (2 )  1 ( ) then we know   i   i  i 

if aii  2 . if aii  2

By proposition 2 and we have   ( )           for definition 2 .

Let  be an imaginary root of g(A) which is symmetrizable GKM algebra . Wecall  a special imaginary root, if  satisfies the following conditions : (s1) (   )  0 - 86 www.ivypub.org/mc

(s2)   ()     (re )  re    (im )  im  (s3)   preserves root multiplicities.

It’s clear that if  is a special imaginary root, then    W , where W  { |  W } .

 2 a 2a  Theorem 3: Let A   b 2 2   where a b  N , and   k11  k22  k33  K  im    b 1 2    k1 k2  k3  N .Then    W if and only if k3  34 k2  32 bk1 k1  N

Proof: () Suppose k3  34 k2  32 bk1 k1  N Then  

k1 2

(21  4 2  33 ) , We have (   )  k12 (2b  5ab2 )  0 .

Use the definition of reflection of imaginary root, we have

 

2 2  1 ( )   (   2a   3a  ) . (  ) k1 (2  5ab)

Thus

  (1 )  1      1    1  4b 2  3b 3 ( 2 3 )2 (1 )   ( 2 )   2       2    2 ( 2 3 )2 ( 2 )   ( 3 )   3       3    3 ( 2 3 )2 (3 ) Therefore    ( 2 3 )2  Since detA  4  10ab  0, we have dim   =3 and   {1 2  3} is a basis of   .

Thus    ( 2 3 )2  W as a reflection on   .

Conversely let   k11  k2 2  k33  K  im ()  2k1  ak1  k3  k3  ak1  k2 .

.

We

know

that

k1 k2  k3  N

with

Then (  )  2 A  0 where A  bk12  ak22  2ak32  abk1k2  abk1k3  2ak1k3 . Using the definition of reflection of imaginary root, we have   (1 )  1  1A (2bk1  abk2  2abk3 ) ,

  ( 2 )   2  1A (abk1  2ak2  2ak3 )

(1)

  (3 )  3  A2 (abk1  ak2  ak3 ) .

(2)

We assert that if    W , then    ( 2 3 )2  If this is not true, for example     3 then   ( 2 ) from the equation (1), we have   3 (2 )  2  3 (abk1  2ak2  2ak3 )k11  (abk1k2  2ak22  2ak2 k3  2 A)2  (abk1k3  2ak2 k3  2ak32  A)3  0

As we know that   {1 2 3} is independent, so we get  (abk1  2ak2  2ak3 )k1  0  2 abk1k2  2ak2  2ak2 k3  2 A  0 2   abk1k3  2ak2 k3  2ak3  A  0

Since k1  0 abk1  2ak2  2ak3  0 and hence A=0, which is a contradiction. Similarly, we can prove that for each - 87 www.ivypub.org/mc

element in -W except ( 2 3 )2 , we get a contradiction.

Suppose    ( 2 3 )2 . Then   ( 2 )  ( 2 3 )2 ( 2 )   2  2   (3 )  ( 2 3 ) (3 )  3

(3) (4)

From the equation (1), (2), (3) and (4), we obtain (abk1  2ak2  2ak3 )  (abk1  ak2  2ak3 )  Therefore k3  34 k2  32 bk1 k1  N complete the proof of theorem 2.

From the theorem 2 above, we obtain the following:

 2 a 2a  k Corollary 1: Let A   b 2 2   where a b  N , and   2 (21  4b 2  3b3 ) k=1,2,... . Then  is a  b 1 2    special imaginary root of g (A).

1

REFERENCES [1]

Bennett.C, “Imaginary roots of a Kac-Moody Lie algebra whose reflections preserve root multiplicities”, Journal of Algebra.

158:244-267, 1993.

[2]

Li Zhenheng, Jin Yidong ,Bai Ruipu, “Relationship between reflections determined by imaginary roots and the Weyl

[3]

Group for a special GKM Algebra”, Journal of Mathematical Research and Exposition, No. 3 Page501-507.1999.

[4]

N. Sthanumoorthy and P. L. Lilly, on some classes of root systems of generalized Kac-Moody algebras.Math.Soc, 2004.

[5]

N. Sthanumoorthy and P. L. Lilly, A note on purely imaginary roots of generalized Kac-Moody algebras. Comm.Algebra, 2003, 31:5467-5480.

[6]

J.E. Humphreys, Introduction to lie algebras and representation theory, Graduate Text of Mathematics, 1972. Springer-Verlag, 9.

[7]

V.G. Kac, Infinite-dimensional Lie algebras, 3rd ed.,1990, Combridge Univ Pres. [3]

[8]

Zhexian Wan, Introduction to Kac-Moody algebra [M]. 1993Beijing:~Science Publishers (in chinese).

AUTHORS 1Xinfang

Song,

received

her

M.S.

2Xiaoxi

Wang is a master student of Beijing University of

degrees in basic mathematics from the

Technology, Beijing, China. Her research interests are

Harbin Normal University, China in

information security, applied mathematics, etc.

2006. She is a Lecturer of Beijing Information Technology college, China.

Her research interests are basic and applied mathematics.

Email: Songxf@bitc.edu.cn.

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