Root multiplicity of a special generalized kac moody algebra eb2

Mathematical Computation September 2014, Volume 3, Issue 3, PP.76-82

Root Multiplicity of a Special Generalized KacMoody Algebra EB2 Xinfang Song1, #, Yinglin Guo2 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: Songxf@bitc.edu.cn

Abstract The author systematically introduces the closed root multiplicity of a special generalized Kac-Moody algebra EB2 which was extended from the finite Kac-Moody algebra B2 .The representative theory is one of the issues of current research in generalized Kac-Moody algebra. The author firstly introduces the basic concepts of the representative theory in detail. Then the multiplicity formula for closed root is popularized from EA2 to EB2. Only through the application of non-homogeneous linear equations can solve the complex calculation of root multiplicity. And then obtain its general solution about the multiplicity of the root for the generalized Kac-Moody algebra EB2. This method has wide applications in other fields.

Keywords: Weight Set; Root System; the Highest Weight Module; Root Multiplicity

1 INTRODUCTION The Kac-Moody algebra has been developing at an accelerating speed. In the past decade the theory had emerged as a field that has close connections with many areas of mathematics and mathematical physics. We know that determination the multiplicities of root for Kac-Moody algebra is one of the issues of current research in generalized Kac-Moody algebra (abbreviated as GKM algebra). The case of finding root multiplicities for affine Kac-Moody algebra has been completely solved. In [4] Berman and Moody derived a closed form root multiplicities formula for all symmetrizable Kac-Moody algebra. In this paper, we get the multiplicities of roots for the GKM algebra EB2 which belongs to a special infinite GKM algebra. As we all know the infinite GKM algebra has complex representative theory. Thus it’s difficult to clearly describe the representative theory for GKM algebra EB2 . From the structure of root we partly research the representative theory for GKM algebra EB2 . And also we get the root multiplicity of a generalized Kac-Moody algebra EB2 Note: The paper also use the symbol from [5] in this paper.

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

Def 2.1[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 aii  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).

Def 2.2 [1] For a GGCM A  (aij )i jI a triple (  ) is a realization of A, where   {1 - 76 www.ivypub.org/MC

  n }    and

  {1 

 n }   satisfying the following:

(1)  is a finite dimensional (complex) vector space such that dim  2n  rankA , and    HomC ( C ) is the cospace of  ; (2)  and   is linearly independent; (3)  i  i  aij denotes a duality pairing between  and   .

Def 2.3 [1] The GKM algebra g(A) associated to a symmetrizable GGCM, A  (aij )i jI is the Lie algebra (over C) generated by the above vector space  and the elements ei  fi (i  I ) satisfying the following relations: (1) [h h  ]  0 , for h、h   ; (2) [h ei ]  hi  ei  [h fi ]    hi  fi 

[ei  f j ]  iji for i,j i, j  I  1 aij

(3) (adei )

1 aij

e j  0 (adfi )

f j  0 if aii  2 and j  i ;

(4) [ei  e j ]  0[ fi  f j ]  0 if aii  a jj  0 and aij  0 .

And the elements of  (respectively   ) are called the simple roots (respectively simple coroots) of g.

Def 2.4 [1] An element   Q is called a root if   0 , g  0 . The number mul  dim g is called the multiplicity of the root  . We believe that muli = mul (i ) = mi for all i  I . A root   0 (resp.   0 ) is called positive (resp. negative). As in the case of Kac-Moody algebras, all the roots are either positive or negative. Let g   g and g   g . Then there’s a triangular decomposition: g  g  g0  g0 .  0

 0

Note 2.1

[1]

 ( Q   Z0i ) is the set of positive roots,  (  ) is the set of negative roots and iI

  

 .

Suppose I re  {i  I | aii  2} ; I im  {i  I | aii  0}

re  {i  | i  I re } , the set of real simple roots and

im  {i  | i  I im } , the set of imaginary simple roots. And let m  (mi  Z | i  I ) be a collection of positive

integers such that mi  1 for all i  I re .

Def 2.5 [1] A g ( A) -modular V is called  -diagonalizable if it admits a weight space decomposition V   V by  weight space V . 

Def 2.6 [1] A nonzero vector from V is defined a weight vector of weight  . Let P(V) = {    V  0} denote the set of weights of V. Def 2.7 [1] The category  is defined as follows: its objects are g(A)-modules V which are  -diagonalizable with finite-dimensional weight spaces so that there exists a finite number of elements 1 P(V ) 

s i 1

 s   such that

D(i )

Def 2.8 [1] A g(A)-modular V is defined a highest-weight module with highest    if there exists a nonzero vector v such that n (v )  0 h (v )  (h)v for h  and U ( g ( A))(v )  V  And the vector v is defined a highest-weight vector.

Def 2.9 [1] A g(A)-module M () with highest weight  is defined a Verma module if every g(A)-module with highest weight  is a quotient of M () .

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Now let P  {   |  (hi )  0 for all i  I ,  (hi )  Z0 if aii  2 }, and let V ( ) be the irreducible highest weight module over g with highest weight  . Denote by T the set of all imaginary simple roots counted with multiplicities and for F  T, we write F   if  (hi ) = 0 for all i  F. Let J be a finite subset of I re and we denote by  J    (  Zi ) J   J   and  ( J )   \ J . jJ

Suppose QJ  Q  (  Zi ) QJ  QJ  Q and Q ( J )   \ QJ . jJ

Define g0( J )    (  g ) , g ( J )   g . ( J )

 (J )

Then there’s a triangular decomposition: g  g( J )  g0( J )  g( J ) , where g0( J ) is the Kac-Moody algebra (with an extended Cartan subalgebra) associated with the generalized Cartan matrix AJ  (aij )i jJ and g ( J ) (resp. g ( J ) ) is a direct sum of irreducible highest weight (resp. lowest weight) modules over g0( J ) .

Suppose WJ   j | j  J  to be the sub group of W generated by the simple reflections  j ( j  J ) , and define W ( J )  { W |      ( J )} . Then WJ is the Weyl group of the Kac-Moody algebra g0( J ) and W (J) be the

set of right coset representatives of WJ in W.

3 THE MAIN RESULTS In this part the paper mainly discusses a special GKM algebra EB2 which was extended from the finite Kac-Moody  0 1 0  algebra B2 . The relative GGCM is A   1 2 2  .  0 1 2    As in the case of symmetrizable Kac-Moody algebra, g ( J ) is viewed as the graded Lie algebra. The paper omits the proof of the following theorem given in [6], [7] in which Kostant’s formula was generalized to GKM algebra to determine the homology modules H k( J ) .

Proposition 31 H k( J ) 



 W ( J )

VJ ( (   s( F ))   )

(3.1)

F T l ( )  F  k

where VJ (  ) denotes the irreducible highest weight module over g0( J ) with highest weight  , and F runs over all the finite subsets of such that any two elements in F are mutually perpendicular. Here we suppose  F  , the number of elements in F and by s(F), the sum of elements in F. Define the homology space H ( J ) of g ( J ) to be H ( J ) 





 W ( J )



 (1)k 1 H k( J )

k 1

VJ ( (   s( F ))   )

F T l ( )  F  k Suppose P( H J )  {  Q ( J ) | dim H( J )  0}  {1 2  3  } and d (i)  dim H( J ) for i = 1, 2, 3 i

For   Q( J ) , suppose T ( J ) ( ) to be the set of all partitions of  into a sum of a sum of  i ’s, that is, - 78 www.ivypub.org/MC

.

T ( J ) ( )  {n  (ni )i 1 | ni  Z0   ni i   } . For n  T J ( ) , the paper will use the notation  n   ni and n   ni .

Now   Q ( J ) is defined to be a function J W    

  

nT

J

 n  1!  

n  d i  .

n!

(3.2)

i



tk and the k 1 k Mobious inversion, a closed form root multiplicity formula for all symmetrizable GKM algebra will be obtained.

The following theorem will be used frequently, the proof of which is given in [7], [8].

The (3.2) will be called the Witt partition function. Then, using the formal power series log 1-t    

Proposition 3.2 Let  be a root of a symmetrizable GKM algebra G. Then there’s

1 J dim g     d W   dad

 da 

1    d  dad



J  a  nT     d 

 n  1! n!

 d i 

ni

Where the classical Mobious function is defined by 1, if n  1;    n   (1) s , if n  if  0, otherwise.

n  n1

ns where ni ( for 1  i  s)are s distinct primes;

Suppose the Borcherds-Cartan matrix A  (aij )i, jI of charge m  (mi  Z 0 | i  I ) satisfies: (1) The set I re is finite; (2) aij  0 for all i, j  I im .

If the paper takes J  I re , the set of all real indices, then the algebra g0( J )    (  g ) is the Kac-Moody algebra  J

(with an extended Cartan subalgebra) associated with the generalized Cartan matrix AJ  (aij )i, jI and the generalized Kac-Moody algebra g  g( J )  g0( J )  g( J ) can be realized as the minimal graded Lie algebra with the local part V  g0( J )  V  , where V im VJ (i ) mi , V  is the contragradient module of V, and mi ' s are the iI

multiplicities of the imaginary simple roots i ' s . In the computation the following proposition is also used, the proof of which is in [9], [10].

Proposition 3.3Suppose that a Borcherds-Cartan matrix A  (aij )i, jI of charge m  (mi  Z0 | i  I ) satisfies: (1) The set I re is finite; (2) aij  0 for all i j  I im .

Let J  I re and consider the corresponding triangular decomposition of the generalized Kac-Moody algebra: g  g ( A m)  g( J )  g0( J )  g( J ) . Then the algebra g ( J ) 

(J )  g (resp. g   g ) is isomorphic to the free

 (J )

 (J )

Lie algebra generated by the space V   VJ (i )m (resp. V    VJ (i )m ), where VJ (  ) (resp. VJ (  ) ) i

iI im

i

iI im

denotes the irreducible highest weight (resp. lowest weight) module over the Kac-Moody algebra g 0J with highest weight  (resp. lowest weight  ).

Remark 31 Proposition 3.1 shows that, under the above assumptions, the generalized Kac-Moody algebra g( J )  g0( J )  g( J ) is isomorphic to the maximal graded Lie algebra with local part V

 g0( J )  V  .

Definition 3.1 An indefinite no-hyperbolic Borcherds-Cartan matrix A is said to be of extended-hyperbolic - 79 www.ivypub.org/MC

Borcherds type if every principal submatrix of A is of finite, affine or hyperbolic Borcherds type.

Now the paper will mainly discuss the root multiplicity formula of extended-hyperbolic generalized Kac-Moody algebra associated with extended-hyperbolic Borcherds-Cartan matrices and determine their root multiplicities explicitly.

The paper mainly uses the results of Kang (cf. [9], [11], [12]).

Theorem3. 1 Regarding the extended-hyperbolic generalized Kac-Moody algebra g  g ( A m) associated with the

 0 1 0  extended-hyperbolic Borcherds-Cartan matrix, A   1 2 2  of charge m  (111) . This matrix is an extension  0 1 2    from finite B2 . Let I  {1 2 3} be the index set for the simple roots of g . And 1 is the imaginary simple root with multiplicity 1 and  2 and  3 are the real simple roots.

Then the Witt partition function becomes:

W ( J ) (l m n) 

There’s have T  {1} since (11 )  0 , F can be either empty or {1} . If it take J  {2 3} then

Proof: g0( J )

1  l   l  k1   l  k1  k2   l  2k1  n      k1 k2  Z    l  k1   k2   k1 k2 n   l  k2  m 

 g0  Ch1 , Where g0  e2  f2  h2  e3  f3  h3  B2 and W ( J )  {1} .

By the formula (3.1), the paper obtain H k( J )  VJ (1 ) and H K( J )  0 for k  2 .

This implies H ( J )  VJ (1 ) , where VJ (1 ) is the 5-dimensional irreducible representation of  (5 C ) .

By identifying l1  m 2  n3  Q with (l m n)  Z0  Z0  Z0 , there’s P( H ( J ) )  {(1 0 0) (11 0) (111) (11 2) (1 21)}

Where (J ) (J ) (J ) (J ) dimH (1( J0)0)  dimH(11  0)  dimH (111)    dimH (11  2)  dimH (121)  1 .

For l m n  Z0 , the only partition of (l , m, n) into a sum of (1,0,0), (1,1,0), (1,1,1), (1,1,2) and (1,2,1) is

l, m, n   a?1  b 2  c 3 d  4  e 5  a(1 0 0)  b(11 0)  c(111)  d (11 2)  e(1 21)

 a  b  c  d  e  l  b  c  d  2e  m  n   c  2d  e Through operation the solution of the linear equation has been gotten:   k2  l  m   k1  k2  n      2k1  k2  m  n    k1     k 2  

k1  k2  Z

Thus the equation (3.3) has been changed to the following:

 l , m, n  =

(k2  l  m) (1,0,0)+ (k1  k2  n) (1,1,0) + (2k1  k2  m  n) (1,1,1) + k1 (1,1,2) + k2 (1,2,1) ,

Then the Witt partition function becomes:

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(3.3)

W ( J ) (l m n) 

1  l   l  k1   l  k1  k2  l  2k1  n       l  k1   k2   k1k2 n   l  k2  m 

(3.4)

Then using the formula (3.3), we obtain the following proposition: Corollary3. 1 Let g  g ( A m) be the extended-hyperbolic GKM algebra associated with the extended-hyperbolic

 0 1 0  Borcherds-Cartan matrix, A   1 2 2  of charge m  (111) . Then, for the root for   a1  b2  c3 with  0 1 2    a b c  Z0 , we have dimg 

 a   a k 1   (d )  kd   kd a d ( abc )  d  d

1

1

2

 a k k  a 2k c   k dk c  a  kd b       d  d  1

1

2

2

1

2

k1 k2  Z 

(3.5)

From Remark 3.1, the algebra g is isomorphic to the maximal graded Lie algebra with local part H ( J )   ((5 C )   )  H ( J ) .

1. If k1  a or k1  k2  a or 2k1  k2  b  c , dim g( abc )  0 .

2. If k1  k2  0 and a = b = c, dim g( aaa) 

1 a

  (d ) . And the subalgebra F0   g(aaa) is the free Lie algebra on a 1

d a

the 1-dimensional space g(111)  .

3. Thus if k1  k2  0 and b = c = 0, we get dim g( a00) 

1 a

  (d ) .

d a

And the subalgebra F0   g(1 0 0) is the free Lie algebra on the 1-dimensional space g(100) . a 1

For example, if k1  k2  0 for the root (8, 4, 2), by the formula (3. 5), it get dim g(8 4 2)  20 .

REFERENCES [1]

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

[2]

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

[3]

Wan Zhexian, Introduction to Kac-Moody algebra [M], scientics publishers (in Chinese).

[4]

S. Berman and R. V. Moody,Multiplicities in Lie algebra, Proc. Amer. Math. Soc. 76 (1979), 223-228.

[5]

N. Sthanumoorthy and P. L. Lilly, on some classes of root systems of generalized Kac-Moody algebras. [J] Math. Soc, vol.343.2004.

[6]

S. Naito, The strong Bernstein-Gelfand-Gelfand resolution for generalized Kac-Moody algebras I: The existence of the resolution, Publ. RIMS, Kyoto Univ. 29(1993), 709-730.

[7]

S.J.Kang and M.H.Kim, Dimension formula for graded Lie algebras and its applications, Trans. Amer.Math. Soc. 351 (1999), 4281-4336.

[8]

N. Sthanumoorthy and P. L. Lillon the root system of generalized Kac-Moody algebra, [J]. J. Madras University, 52(2000), 81-103.

[9]

S.J.Kang and M.H.Kim, DimensionFormula for graded Lie algebras and its applications, Trans. Amer. Math. Soc. 351 (1999), 4281-4336.

[10] E. Jursich, Generalized Kac-Moody algebras, free Lie algebraand the structure of the Monster Lie algebra, J. Pure and Applied algebra 126(1998), 233-266.

[11] S. J. Kang, Root Multiplicities of graded Lie algebras, in: Lie algebras and their representations, S. J. Kang, M. H. Kim, I. S. Lee (eds).

[12] S. J. Kang, generalized Kac-Moogy algebras and the modular functin j, Math. Ann 298(1994), 373-384. Contemp.Math.194 (196), 161-176.

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AUTHORS 1Xinfang

Song received her M.S. degrees

2Yinglin

Guo is a master student of Beijing University of

in basic mathematics from the Harbin

Technology, Beijing, China. His research interests are

Normal University, China in 2006. She is

information security, cryptography, etc.

a

Lecturer

of

Beijing

Information

Technology college, China. Her research interests

are

basic

and

applied

mathematics, information security.

Email: Songxf@bitc.edu.cn.

- 82 www.ivypub.org/MC