Combinatorial Identities Related To Root Supermultiplicities In Some Borcherds Superalgebras

The International Journal Of Engineering And Science (IJES) 

|| Volume || 4 || Issue || 6 || Pages || PP.63-77|| June - 2015 || 

ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805 

Combinatorial Identities Related To Root Supermultiplicities In 

Some Borcherds Superalgebras 

N.Sthanumoorthy * and K.Priyadharsini 

The Ramanujan Institute for Advanced study in Mathematics 

University of Madras, Chennai - 600 005, India. 

-------------------------------------------------------------ABSTRACT--------------------------------------------------------- 

In this paper, root supermultiplicities and corresponding combinatorial identities for the Borcherds 

superalgebras which are extensions of B and B are found out. 

2 

3 

Keywords: Borcherds superalgebras, Colored superalgebras, Root supermultiplicities. 

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Date of Submission: 10-June-2015 

Date of Accepted: 25-June-2015 

---------------------------------------------------------------------------------------------------------------------------------------- 

I. INTRODUCTION 

The theory of Lie superalgebras was constructed by Kac in 1977. The theory of Lie superalgebras can 

also be seen in Scheunert(1979) in a detailed manner. The notion of Kac-Moody superalgebras was introduced 

by Kac(1978) and therein the Weyl-Kac character formula for the irreducible highest weight modules with 

dominant integral highest weight which yields a denominator identity when applied to 1-dimensional 

representation was also derived. Borcherds(1988, 1992) proved a character formula called Weyl-Borcherds 

formula which yields a denominator identity for a generalized Kac-Moody algebra.Kang developed homological 

theory for the graded Lie algebras in 1993a and derived a closed form root multiplicity formula for all 

symmetrizable generalized Kac-Moody algebras in 1994a. Miyamoto(1996) introduced the theory of 

generalized Lie superalgebra version of the generalized Kac-Moody algebras(Borcherds algebras). Kim and 

Shin(1999) derived a recursive dimension formula for all graded Lie algebras. Kang and Kim(1999) computed 

the dimension formula for graded Lie algebras. Computation of root multiplicities of many Kac-Moody algebras 

and generalized Kac-Moody algebras can be seen in Frenkel and Kac(1980), Feingold and Frenkel(1983),Kass 

et al.(1990), Kang(1993b, 1994b,c,1996), Kac and Wakimoto(1994), Sthanumoorthy and Uma 

Maheswari(1996b), Hontz and Misra(2002), Sthanumoorthy et al.(2004a,b) and Sthanumoorthy and 

Lilly(2007b). Computation of root multiplicities of Borcherds superalgebras was found in Sthanumoorthy et 

al.(2009a). Some properties of different classes of root systems and their classifications for Kac-Moody algebras 

and Borcherds Kac-Moody algebras were studied in Sthanumoorthy and Uma Maheswari(1996a) and 

Sthanumoothy and Lilly(2000, 2002,2003,2004, 2007a). Also, properties of different root systems and complete 

classifications of special, strictly, purely imaginary roots of Borcherds Kac-Moody Lie superalgebras which are 

extensions of Kac-Moody Lie algebras were explained in Sthanumoorthy et al.(2007,2009b) and Sthanumoorthy 

and Priyadharsini(2012, 2013). Moreover, Kang(1998) obtained a superdimension formula for the homogeneous 

subspaces of the graded Lie superalgebras, which enabled one to study the structure of the graded Lie 

superalgebras in a unified way. Using the Weyl-Kac-Borcherds formula and the denominator identity for the 

Borcherds superalgerbas, Kang and Kim(1997) derived a dimension formula and combinatorial identities for the 

Borcherds superalgebras and found out the root multiplicities for Monstrous Lie superalgebras. Borcherds 

superalgebras which are extensions of Kac Moody algebras A and A were considered in Sthanumoorthy et 

2 

3 

al.(2009a) and therein dimension formulas were found out.In Sthanumoorthy and Priyadharsini(2014), we have 

computed combinatorial identities for A and A and root super multiplicities for some hyperbolic Borcherds 

2 

3 

superalgebras. 

The aim of this paper is to compute dimensional formulae, root supermultiplicities and 

corresponding combinatorial identities for the Borcherds superalgebras which are extensions of Kac-Moody 

algebras B and B . 

2 

3 

www.theijes.com The IJES Page 63

Combinatorial Identities Related To Root… 

II. PRELIMINARIES 

In this section, we give some basic concepts of Borcherds superalgebras as in Kang and Kim (1997). 

Definition 2.1: Let I be a countable (possibly infinite) index set. A real square matrix 

Borcherds-Cartan matrix if it satisfies: 

(1) a = 2 or a 0 for all i I , 

ii 

ii 

(2) a 0 if i j and a Z if a = 2 , 

ij 

ij 

ii 

A 

= ( ) is called 

a ij 

i , j 

I 

(3) a 0 a = 0 

ij 

= 

ji 

We say that an index i is real if a = 2 and imaginary if a 0 . We denote by 

I 

ii 

re 

= { i I 

| a 

ii 

= 

2}, 

I 

ii 

im 

= { i I 

Let m { m Z | i } be a collection of positive integer such that m = 1 for all i I 

i 

= I 

> 0 

| a 

ii 

i 

 

0}. 

re 

. We call m a 

charge of A . 

Definition 2.2: A Borcherds-Cartan matrix A is said to be symmetrizable if there exists a diagonal matrix 

D = d iag ( ; i I ) with > 0 ( i I ) such that 

i 

DA is symmetric. 

Let 

C 

for all 

c ij 

i , j 

I 

i 

= ( ) be a complex matrix satisfying c = 1 for all i , j I . Therefore, we have c = 1 

c ij ji 

i I . We call i I an even index if c = 1 and an odd index if c = 1 . 

We denote by 

ven 

I e o dd 

( I ) the set of all even (odd) indices. 

Definition 2.3: A Borcherds-cartan matrix 

satisfies: 

A 

ii 

a ij 

i , j 

I 

ii 

= ( ) is restricted (or colored) with respect to C if it 

If a = 2 and c = 1 then a are even integers for all j I . In this case, the matrix C is called a 

ij 

ii 

coloring matrix of A . 

ii 

Let h = ( C h ) ( C d ) be a complex vector space with a basis { h , d ; i I } , and for each 

i 

I i 

i 

I i 

i i 

i I , define a linear functional h * by 

i 

( h ) = a , ( d ) = for all j I ........... 

i 

j 

ji 

i 

j 

ij 

If A is symmetrizable, then there exists a symmetric bilinear form (|) on h * satisfying 

( | ) = a = a for all i, j I . 

i 

j 

Definition 2.4: Let Q 

i 

ij 

j 

ji 

= Z and i 

I i 

Q = 

i 

I 0 i 

......( 

2 .1) 

Z , Q = Q . Q is called the root lattice. 

 

 

The root lattice Q becomes a (partially) ordered set by putting if and only if Q . 

 

The coloring matrix C 

X 

= ( ) defines a bimultiplicative form : Q Q C by 

c ij 

i , j 

I 

( , ) = for 

alli , j I , 

i 

j 

c ij 

( , ) = ( , ) ( , ), 

( , ) = ( , ) ( , ) 

for all , , Q . Note that satisfies 

( , ) ( , ) = 1 forall 

, Q ,.......... ....( 2 .2 ) 

since c = 1 for all i , j I . In particular ( , ) = 1 for all Q . 

c ij ji 

We say Q is even if ( , ) = 1 and odd if ( , ) = 1 . 

Definition 2.5: A -colored Lie super algebra is a Q -graded vector space 

bilinear product [, ] : L L L satisfying 

L 

L L , 

, 

 

x 

, y = ( , )[ y , x ], 

L = L together with a 

Q 

 

ii 


x 

, y , z = [[ x , y ], z ] ( , )[ y , [ x , z ]] 

for all , Q and x L y L , z L 

In a -colored Lie super algebra 

[ x , [ x , x ]] = 0 if is odd. 

, . 

 

L 

= L , for 

Q 

 


x L , we have [ x , x ] = 0 if is even and 

Definition 2.6: The universal enveloping algebra U ( L ) of a -colored Lie super algebra L is defined to be 

T ( L )/ J , where T ( L ) is the tensor algebra of L and J is the ideal of T ( L ) generated by the elements 

[ x , y ] x y ( , ) y x x L , y L ) . 

( 

 

 

Definition 2.7: The Borcherds superalgebra g = g ( A , m , C ) associated with the symmetrizable Borcherds- 

Cartan matrix A of charge m = ( m ; i I ) and the coloring matrix C = ( c 

i ij 

) is the -colored Lie 

i , j 

I 

super algebra generated by the elements h d ( i I ), e , f ( i I , k = 1,2, , m ) with defining 

relations: 

h 

, h = h 

, d = d 

, d 

i j 

i j 

i j 

h 

, e = a e , h 

, f = 

i jl 

ij jl i jl 

d 

, e = e , d 

, f = 

i jl 

ij jl i jl 

e 

, f = h 

ik 

( ade 

) 

1 a 

ij 

for i , j I , k = 1, , m , l = 1, , m . 

= 

, i i 

ik ik 

i 

( adf 

e 

, e = f , f = 0 if a = 0 

i 

ik 

ik 

jl 

jl 

e 

jl 

j 

ij 

ik 

kl 

ik 

) 

i 

1 a 

ij 

jl 

f 

jl 

ij 

= 

a 

 

ij 

0, 

ij 

f 

f 

jl 

= 0 if a 

The abelian subalgebra h ( C h ) ( C d ) is called the Cartan subalgebra of g and the linear 

= i 

I i 

i 

I i 

functionals h * ( i I ) defined by (2.1) are called the simple roots of g . For each 

i 

r i 

GL (h* ) be the simple reflection of h * defined by 

r ( ) = ( ) ( h* ). 

i 

h i 

i 

jl 

ii 

, 

, 

= 

2 

and 

i 

j , 

i I 

re 

The subgroup W of GL (h* ) generated by the r ’s ( i I ) is called the Weyl group of the Borcherds super 

i 

algebra g . 

The Borcherds superalgebra g = g ( A , m , C ) has the root space decomposition g = g , where 

Q 

Note that 

and 

g 

 

= { x g | [ h , x ] = ( h ) x for all h h}. 

g 

g 

 

i 

 

i 

= C 

C 

e e 

i ,1 

i , m 

i 

= C 

C 

f f 

i ,1 

i , m 

i 

X 

We say that Q is a root of g if g 0 . The subspace g is called the root space of g attached to 

 

 

. A root is called real if ( | ) > 0 and imaginary if ( | ) 0 . 

re 

im 

In particular, a simple root is real if a = 2 that is if i I and imaginary if a 0 that is if i I . 

i 

ii 

Note that the imaginary simple roots may have multiplicity > 1 . A root > 0 ( < 0 ) is called positive 

(negative). One can show that all the roots are either positive or negative. We denote by 

of all roots, positive roots and negative roots, respectively. Also we denote 

roots of g . Define the subspaces g 

 

= g . 

 

 

 

 

Then we have the triangular decomposition of g : g = g h g . 

 

( 

0 

1 

ii 

, and 

 

 

re 

, let 

the set 

) the set of all even (odd) 



Definition 2.8:(Sthanumoorthy et al.(2009b)) We define an indefinite nonhyperbolic Borcherds - Cartan 

matrix A, to be of extended-hyperbolic type, if every principal submatrix of A is of finite, affine, or hyperbolic 

type Borcherds - Cartan matrix. We say that the Borcherds superalgebra associated with a Borcherds - Cartan 

matrix A is of extended-hyperbolic type, if A is of extended-hyperbolic type. 

Definition 2.9: A g -module V is called 

h -diagonalizable, if it admits a weight space decomposition 

V 

 

= V 

h 

, 

where V = v 

V | h v = ( h ) vfor all h h. 

 

If V 0, then 

 

is called a weight of V , and dim V is called the multiplicity of 

 

in V . 

Definition 2.10: A h -diagonalizable g -module V is called a highest weight module with highest weight 

 

h , if there is a nonzero vector v V such that 

 

1. e ik 

v 

= 0, for all i I , k = 1, , m , 

i 

2. h v = ( h ) v for all h h and 

 

3. V = U ( g ) v . 

 

The vector v is called a highest weight vector. 

 

For a highest weight module V with highest weight , we have 

(i) V 

(ii) V 

(iii) dim 

 

= U ( g ) v , 

= 

 

 

 

V , V = C v and 

 

V < for all . 

 

 

Definition 2.11: Let P (V ) denote the set of all weights of V . When all the weights spaces are finite 

 

dimensional, the character of V is defined to be chV 

= (dim V ) e , 

where 

 

h 

* 

 

e are the basis elements of the group C [h* ] with the multiplication given by 

 

 

e e = e for 

, h * . Let b = h g be the Borel subalgebra of g and C be the 1-dimensional b -module 

 

 

 

 

defined by g 1 = 0, h 1 = ( h )1 for all h h. 

The induced module M ( ) = U ( g ) C 

 

U ( b 

is 

) 

called the Verma module over g with highest weight . Every highest weight g -module with highest weight 

is a homomorphic image of M ( ) and the Verma module M ( ) contains a unique maximal submodule 

J ( ) . Hence the quotient V ( ) = M ( )/ J ( ) is irreducible. 

Let 

 

P be the set of all linear functionals h * satisfying 

The elements of 

( h 

 

( h 

 

( h 

 

 

i 

i 

i 

) Z 

0 

) 2 Z 

) 0 

0 

 

P are called the dominant integral weights. 

re 

for 

all i I 

re 

for 

all i I 

im 

for 

all i I 

1 

Let h * be the C -linear functional satisfying ( h i 

) = a for all i I 

ii 

2 

 

I 

o dd 

. Let T denote the set of all 

imaginary simple roots counted with multiplicities, and for F T , we write F , if h ) = 0 for all 

F . 

i 

( 

i 

Definition 2.12:[Kang and Kim (1997)] Let 

J be a finite subset of 

e 

I r 

. We denote by 



 

J 

 

), = 

j 

J j J 

 

 

 

= ( Z and ( J ) = \ . Let 

J 

J 

and 

Then 

g 

( J ) 

0 

= h 

g 

( J ) 

 

 

= 

 

 

 

 

 

 

J 

 

g 

 

( J ) 

 

g 

 

, 

(2.3) 

 

 

( J ) 

g is the restricted Kac-Moody super algebra (with an extended Cartan subalgebra) associated with the 

0 

Cartan matrix 

A 

= ( ) and the set of odd indices 

J 

a ij 

i , j 

J 

= { j J | c = 1} 

jj 

J 

o dd 

 

. 

= J 

( J ) ( J ) ( J ) 

Then the triangular decomposition of g is given by g = g g g . 

Let W 

= r j J be the subgroup of W generated by the simple reflections r ( j J ), 

j 

and let 

| 

J j 

 

 

 

0 

I 

o dd 

 

W ( J ) = w W | ( J 

1 

where = { | w < 0}.(2.4) 

Therefore 

w 

W is the Weyl group of the restricted Kac-Moody super algebra 

J 

w 

 

) 

 

( J ) 

g and W ( J ) is the set of right 

coset representatives of W in W . That is W = W W ( J ). 

J 

J 

The following lemma given in Kang and Kim (1999), proved in Liu (1992), is very useful in actual computation 

of the elements of W(J). 

0 

Lemma 2.13: Suppose 

w 

= w r 

and l ( w ) = l ( w ) 

1. Then w W ( J ) if and only if w W ( J ) and 

j 

w ( ) ( J ). 

J 

 

 

 

Let = ( i = 0,1) 

i , J 

i 

J 

 

 

 

and ( J ) = \ ( i = 0,1). Here ( ) denotes the set of all 

i 

i 

positive or negative even (resp., positive or negative odd) roots of g . 

The following proposition, proved in Kang and Kim (1997), gives the denominator identity for Borcherds 

superalgebras. 

Proposition 2.14: [Kang and Kim (1997)]. Let J be a finite subset of the set of all real indices 

 

 

 

0 ( J ) 

 

 

 

1 ( J ) 

(1 e 

(1 e 

dim g 

 

) 

dim g 

 

) 

= 

 

wW 

( J ) 

F T 

i , J 

( 1) 

l ( w ) | F | 

c hV 

J 

0 

1 

e 

I r 

( w ( s ( F ) )) 

. Then 

where V ( ) denotes the irreducible highest weight module over the restricted Kac-Moody super algebra 

J 

( J ) 

g with highest weight 

0 

and where F runs over all the finite subsets of T such that any two elements of 

F are mutually perpendicular. Here l ( w ) denotes the length of w , | F | the number of elements in F , and 

s ( F ) the sum of the elements in F . 

å 

 

 

Definition 2.15: A basis elements of the group algebra C [ h ] by defining E = ( , ) e . 

Also define the super dimension Dim g of the root space g by 

 

 

Dim 

g 

 

= ( , ) dim g .......... 

 

.......... 

....... 

(2.5) 

Since w ( s ( F )) is an element of Q , all the weights of the irreducible highest weight g 

 

V J 

( w ( s ( F )) ) are also elements of Q . 

 

www.theijes.com The IJES Page 67 

( J ) 

0 

-module

Hence one can define the super dimension 


DimV of the weight space V of V ( w ( s ( F )) ) 

 

 

J 

in a 

( ) 

similar way. More generally, for an h -diagonalizable g J 

-module V = V such that P ( V ) Q , we 

0 

å 

define the super dimension 

For each k 1 , let 

H 

( J ) 

and define the homology space 

k 

DimV of the weight space V to be 

 

 

= 

H 

wW 

( J ) 

( J ) 

 

F T 

( J ) 

H of 

= 

 

 

k = 1 

DimV 

l ( w ) | F |= k 

 

V 

J 

( J ) 

g to be 

 

( 1) 

an alternating direct sum of the vector spaces. 

For 

Q 

 

, define the super dimension 

DimH 

= 

 

 

k = 1 

= 

k 1 

H 

= ( , ) dim V (2.6) 

 

h 

( w ( s ( F )) ).......... 

( J ) 

k 

= 

H 

( J ) 

1 

H 

( J ) 

2 

 

H 

( J ) 

( J ) 

DimH 

of the -weight space of 

 

( J ) 

k 1 

( J ) 

 

( 1) 

( DimH 

) 

k 

( 1) 

k 1 

k = 1 

wW 

( J ) 

 

F T 

DimV 

J 

3 

.......... ...... (2.7) 

 

, (2.8) 

( J ) 

H to be 

( w ( s ( F )) ) 

 

l ( w ) | F |= k 

= 

wW 

( J ) 

 

( 1) 

l ( w ) | F | 1 

DimV J 

( w ( 

s ( F )) ) .......... ........( 

 

2.9) 

F T 

Let 

P ( H 

( J ) 

) = 

l ( w ) | F | 1 

 

( J ) 

 

Q ( J ) | dim H 0 .......... ......... (2.10) 

( J ) 

and let , , , be an enumeration of the set P ( H ) . Let 

, 

1 2 3 

 

( J ) 

D ( i) 

= DimH . 

 

i 

) 

Remark: The elements of P ( H 

( J 

) can be determined by applying the following proposition, proved in Kac 

(1990). 

Proposition 2.16: (Kang and Kim, 1997) 

Let P . Then P ( ) = W .{ P | • is nondegener ate with respect to } . 

 

 

Now for Q 

( J ), one can define 

T 

( J ) 

n 

= ( n ) | n Z , n = ,.......... .......... ...(2.11) 

i i 1 

i i i 

( ) = 

0 

( J ) 

which is the set of all partitions of into a sum of ’s. For n T ( ), use the notations | n |= 

i 

n and 

i 

n = n ! . 

! 

i 

Now, for Q 

( J ) 

( J ), the Witt partition function W ( ) is defined as 

W 

( ) = 

 

( J ) 

nT 

( ) 

(| n | 1)! 

( J ) 

n 

i 

n! 

 

D ( i ) 

........... .......... ..(2.12) 



Now a closed form formula for the super dimension 

the following theorem. The proof is given in Kang and Kim(1997). 

Theorem 2.17: Let J be a finite subset of 

= 

 

d | 

Dim 

1 

( d ) 

d 

g 

 

 

= 

( J ) 

nT 

 

d 

 

Dim of the root space g ( ( J )) is given by 

g 

 

e 

I r 

. Then, for ( J ) , we have 

 

d | 

1 

( d ) W 

d 

(| n | 1)! 

n! 

 

( J ) 

 

 

 

 

d 

D ( i) 

n 

i 

 

 

 

.......... .(2.13) 

where is the classical Möbius function. Namely, for a natural number n , ( n ) is defined as follows: 

1 

for 

n = 1, 

k 

( n ) = ( 1) for 

n = p p ( p , 

k 1 

 

0 if 

it is not square free 

1 

 

, p 

k 

 

: d istinct 

primes 

 

and, for a positive integer d , d | denotes = d 

for some Q , in which case = . 

 

d 

In the following sections 3.1 and 3.2, we find root supermultiplicities of Borcherds superalgebras which are the 

Extensions of Kac-Moody Algebras B and B (with multiplicity 

2 

3 

1 ) and the corresponding combinatorial 

identities using Kang and Kim(1997). 

), 

III. 

Root supermultiplicities of some Borcherds superalgebras which are the Extensions of Kac 

Moody Algebras and the corresponding combinatorial identities 

3.1 Superdimension formula and the corresponding combinatorial identity for the extended-hyperbolic 

Borcherds superalgebra which is an extension of B 

2 

Below, we find the dimension formulae and combinatorial identities for the Borcherds superalgebras which are 

extension of B . (for a same set 

2 

re 

( J ) 

J ) by solving T ( ) in two different method. 

Consider the extended-hyperbolic Borcherds superalgebra g = g ( A , m , C ) associated with the extendedhyperbolic 

Borcherds-Cartan super matrix, 

k a b 

1 

 

 

 

1 

A = a 2 1 and the corresponding coloring matrix C= c 

1 

 

 

b 2 2 

1 

 

c 

2 

c 

c 

1 

1 

3 

1 

c 

c 

1 

2 

3 

 

 

 

 

 

X 

with c c , c C . 

, 

1 2 3 

Let I = {1,2,3} be the index set for the simple roots of g . Here is the imaginary odd simple root with 

1 

multiplicity 1 and , are the real even simple roots. 

2 3 

1 

Let us consider the root k k k . We have ( , ) = ( 1) . Hence is an even 

= Q 

1 1 2 2 3 3 

root(resp. odd root) if k is even integer(resp. odd integer). Also T = { } and the subset 

1 

1 

F T is either 

re 

empty or { 1 

}. Take J as J = {3}. By Lemma 2.13, this implies that W ( J ) = {1, r }. From the 

2 

equations(2.7) and (2.8), the homology space can be written as 

H 

H 

H 

( J ) 

 

1 

= V ( 

) V ( ) 

J 1 

J 2 

( J ) 

 

2 

( J ) 

k 

= V ( 

( a 1) ) 

J 1 

2 

= 

0 

k 3 

( J ) 

( J ) 

Therefore H = H H 

1 

( J ) 

2 

= V ( 

) V ( 

) V ( 

( a 1) ) 

J 1 J 2 J 1 

2 

2 

k 



( J ) 

( J ) 

with DimH = 1; DimH = 1 

 

(1,0,0) (0,1,0) 

DimH 

( J ) 

( J ) 

= 1; DimH = 1. 

(1,1,1) (1,1, a 1) 

( J ) 

We take P ( H ) = {(1,0,0), (0,1,0), (1,1,1), (1,1, a 1)}. 

Let = p q u Q , with p , q , t ) Z Z Z . Then by proposition.(2.16) 

= 

1 2 

3 

( 0 0 0 

( ) 

we get T J ) = {( s , s , s , s ) | s (1,0,0) s (0,1,0) s (1,1,1) s (1,1, a 1) = ( p , q , u )}. 

This implies 

( 1 2 3 4 1 

2 

3 

4 

s s s = 

 

1 3 

s s s = 

 

2 3 

4 

4 

p 

q 

s a 1) s = 

( 

3 4 

u 

We have s = p u as 

1 4 

s = q u as 

2 

s = u ( a 1) s 

3 

s 

4 

= 0 to 

4 

u 

min ( p , q , [ 

a 

Applying s , s , s , s in Witt partition formula(eqn.(2.12)), we have, 

1 2 3 4 

W 

( J ) 

( ) = 

4 

u 

min ( p , q ,[ ]) 

a 1 

 

s = 0 

4 

From eqn(2.13), the dimension of 

= 

 

d | 

Dim 

1 

( d ) 

d 

g is 

 

g 

 

 

]) 

1 

( p u q as 

( p q )! ( q u as 

= 

( J ) 

nT 

 

d 

 

d | 

1 

( d ) W 

d 

(| n | 1)! 

n! 

 

( J ) 

4 

1)! ( 1) 

)! ( u ( a 1) s 

4 

 

 

 

 

d 

D ( i ) 

n 

i 

 

 

 

, 

p u q as 

4 

4 

)! s 

4 

.......... ...(3.1.1) 

! 

( J ) 

Substituting the value of W ( ) from eqn. (3.1.1) in the above dimension formula, we have, 

Dim 

g 

= 

 

d | 

1 

d 

( d ) 

p q t 

min ( , ,[ ]) 

d d d ( a 1) 

 

s = 0 

4 

p 

( 

d 

p 

( 

d 

 

q 

d 

u 

d 

)! ( 

q 

d 

 

 

q 

d 

u 

d 

 

 

a 

d 

a 

d 

s 

4 

s 

1)! ( 1) 

4 

)! ( 

p / d u / d q / d as / d 

4 

u 

 

( a 1) 

s 

d d 

4 

)! ( 

4 

. 

s / d )! 

.......... ..( 3 .1 .2 ) 

If we solve the same ( 

( J ) 

( J ) 

P H ), using partition and substituting this partition in T ( ) , we have 

T 

( J ) 

( ) = {( p q ), u , u ( a 1) , }, 

1 

2 2 

where is the partition of ` u with parts (1, a 1) of length ‘u’ and is the partition of s with parts upto 

1 

2 

4 

min ( p , q , [ 

u 

( J ) 

]) . Applying T ( ) in Witt partition formula(eqn.(2.12)), we have 

a 1 

W 

( J ) 

( ) = 

 

( J ) 

, T 

( ) 

1 2 

From eqn.(2.13), the dimension of 

g is 

 

1 

( p u 1)! ( 1) 

1 

.......... ........(3 

( p q )! ( u )! ( u ( a 1) )! ! 

1 

p u 


2 

2 

.1.3)


= 

Dim 

 

d | 

g 

 

= 

1 

( d ) 

d 

 

d | 

1 

( d ) W 

d 

 

( J ) 

nT 

 

d 

( J ) 

n! 

 

 

 

 

d 

(| n | 1)! 

 

 

 

 

D ( i) 

( J ) 

Substituting the value of W ( ) from eqn. (3.1.2) in the above dimension formula, we have 

n 

i 

, 

Dim 

g 

 

= 

 

d | 

1 

( d ) 

d 

 

 

( J ) ( J ) 

nT 

 

, T 

( ) 

1 2 

d 

Which gives another form of dimension of 

Applying the value of s 4 in (3.1.1), we get 

p 

( 

d 

g 

. 

 

q 

d 

p 

( 

d 

)! ( 

 

d 

1 

u 

d 

 

 

1 

 

d 

u 

d 

)! ( 

1)! ( 1) 

p / d u / d / d 

1 

u 

( a 1)( / d ))! ( / d )! 

2 

2 

d 

W 

( J ) 

( ) 

= 

u 

min ( p , q ,[ ]) 

a 1 

 

s = 0 

4 

( p u q as 

( p q )! ( q u as 

4 

1)! ( 1) 

)! ( u ( a 1) s 

4 

p u q as 

4 

4 

)! s 

4 

! 

 

( p u q 1)! ( 1) 

( p q )! ( q u 

p u q 

)! ( u )! 

 

( p u q a 1)! ( 1) 

p u q a 

( p q )! ( q u a )! ( u ( a 1))!1 

....... upto 

! 

min( 

p , q , [ u / a 

1]) 

= 

 

( J ) 

, T 

( ) 

1 2 

1 

( p u 1)! ( 1) 

1 

. 

( p q )! ( u )! ( u ( a 1) )! ! 

1 

p u 

where is the partition of ` u with parts (1, a ) of length ‘ u as ’ and is the partition of s with parts 

1 

4 

2 

4 

upto min ( p , q , [ 

u 

]) . 

a 1 

Hence, we get the following theorem. 

Theorem 3.1.1: For the extended-hyperbolic Borcherds superalgebra g = g ( A , m , C ) associated with the 

extended-hyperbolic Borcherds-Cartan super matrix 

consider the root 

= p q u Q . Then the dimension of g is 

1 2 

3 

 

2 

2 

k a b 

 

 

A = a 2 1 with charge m = {1,1,1}, 

 

 

b 2 2 

Dim 

g 

 

= 

 

d | 

1 

d 

( d ) 

p q t 

min ( , ,[ ]) 

d d d ( a 1) 

 

s = 0 

4 

p 

( 

d 

p 

( 

d 

 

q 

d 

u 

d 

)! ( 

q 

d 

 

 

q 

d 

u 

d 

 

 

a 

d 

a 

d 

s 

4 

s 

1)! ( 1) 

4 

)! ( 

p / d u / d q / d as / d 

4 

u 

 

( a 1) 

s 

d d 

4 

)! ( 

4 

. 

s / d )! 

Moreover the following combinatorial identity holds: 



u 

min ( p , q ,[ ]) 

a 1 

 

s = 0 

4 

( p u q as 

( p q )! ( q u as 

4 

1)! ( 1) 

)! ( u ( a 1) s 

4 

p u q as 

4 

4 

)! s 

4 

= 

! 

 

( J ) 

, T 

( ) 

1 2 

( p 

1 

( p u 1)! ( 1) 

1 

.....(3.1. 

q )! ( u )! ( u ( a 1) )! ! 

1 

p u 

2 

2 

4) 

where is the partition of ` u with parts (1, a ) of length ‘ u as ’ and is the partition of s with parts 

1 

4 

2 

4 

upto min ( p , q , [ 

u 

]) . 

a 1 

k 1 b 

 

 

Example 3.1.2: For the Borcherds-Cartan super matrix A = 1 2 1 , 

consider a root 

 

2 2 

 

b 

= = (5,4,3) with a=1. 

Substituting = = (5,4,3), a = 1 in eqn(3.1.1), we have 

W 

( J ) 

( ) = 

= 

u 

min ( p , q ,[ ]) 

a 1 

 

s = 0 

4 

3 

min (5,4, [ ]) 

2 

 

s = 0 

4 

( p u q as 

( p q )! ( q u as 

(5 3 4 s 

(5 4)! (4 3 s 

1)! ( 1) 

)! ( u ( a 1) s 


4 

4 

4 

1)! ( 1) 

4 

)! (3 

5 3 4 s 

4 

2 s 

= 

5! ( 1) 

 

7! ( 1) 

= 4.5 3.4.5.6.7 

1!1!3! 1!2!1!1! 

Substituting = = (5,4,3), a = 1 in eqn(3.1.3), we have 

W 

( J ) 

( ) = 

 

6 

( J ) 

, T 

( ) 

1 2 

5! ( 1) 

= 

1!1!3! 

Hence the equality (3.1.4) holds. 

6 

7 

( p u 1)! ( 1) 

4 

)! s 

p u q as 

4 

4 

! 

4 

= 2500. 

( p q )! ( u )! ( u ( a 1) )! ! 

7! ( 1) 

 

1!2!1!1! 

7 

1 

1 

= 2500. 

p u 

1 

3.2.Dimension Formula and combinatorial identity for the Borcherds superalgebra which is an extension 

of B 

3 

Here we are finding the superdimension Formula and combinatorial identity for the Borcherds superalgebra 

which is an extension of B using the same 

3 

J 

2 

2 

)! s 

re 

( J ) 

and solving T ( ) in two different ways. 

Consider the extended-hyperbolic Borcherds superalgebra g = g ( A , m , C ) associated with the extendedhyperbolic 

Borcherds-Cartan super matrix 

1 

 

1 

c 

1 

matrix C= 

1 

c 

2 

 

1 

c 

3 

c 

c 

c 

1 

1 

4 

1 

5 

1 

c 

c 

c 

1 

1 

6 

2 

4 

c 

c 

c 

1 

3 

5 

6 

 

 

 

 

 

 

k a b c 

 

 

a 2 1 0 

A = 

and the corresponding coloring 

 

b 1 2 1 

 

 

 

c 0 2 2 

X 

with c c , c , c C . 

, 1 2 3 4 

4 

!


Let I = {1,2,3,4} be the index set with charge m = {1,1,1,1}. 

1 

Let us consider the root k k k k . We have ( , ) = ( 1) . Hence is 

= Q 

1 1 2 2 3 3 4 4 

an even root(resp. odd root) if k is even integer(resp. odd integer). Also T = { } and the subset 

1 

1 

F T is 

re 

either empty or { 1 

}. Take J as J = {2,3}. By Lemma 2.13, this implies that W ( J ) = {1, r }. 

4 

From the equations (2.7) and (2.8),the homological space can be written as 

( J ) 

H = V (1( ) ) V ( r ( ) ) 

1 J 

1 

J 4 

= V ( 

) V ( 

) 

J 1 J 4 

H 

2 J 

H 

( J ) = V ( 

( c 1) ) 

1 

4 

( J ) 

k 

= 

0 

k 3 

2 

k 

with 

ThereforeH 

DimH 

( J ) 

( J ) 

( J ) 

= H = V ( 

) V ( 

)! 

V ( 

( c 1) ) 

1 

J 1 J 4 J 1 

4 

( J ) 

( J ) 

= 1; DimH = 1; DimH = 1; 

(1,0,0,0) (1,1,0,0) 

(1,1,1,0) 

Hence we have 

DimH 

P ( H 

( J ) 

( J ) 

= 1; DimH = 1 

(1,1,1,1) (1,1,1, c 1) 

( J ) 

) = {(1,0,0,0) 

, (1,1,0,0), 

(1,1,1,0), 

(1,1,1,1), 

(1,1,1, 

c 1)}. 

Let = p q u v Q , with p , q , u , v ) Z Z Z Z . Then by 

= 1 2 

3 

4 

proposition.(2.16), we get 

( 0 0 0 0 

( 

) 

T J ( ) = {( s , s , s , s , s ) | s (1,0,0,0) s (1,1,0,0) s (1,1,1,0) 

1 2 3 4 5 1 

2 

3 

This implies 

s (1,1,1,1) s (1,1,1, c 1) = ( p , q , u , v )}. 

4 5 

s s s s = 

 

1 2 3 

s s s s = 

 

2 3 4 

s s s = 

 

3 4 

5 

4 

u 

s ( c 1) s = v. 

4 5 

5 

p 

q 

We have s = p q v cs 

1 5 

s 

2 

= 

q 

u 

s = u v cs 

3 

5 

s = q p s 

4 5 

s 

5 

= 0 to 

v 

min ( p , q , u , [ 

c 

]). 

1 

Applying s , s , s , s , s in Witt partition formula(eqn.2.12), we have 

1 2 3 4 5 

W 

( J ) 

( ) = 

v 

min ( p , q , u ,[ ]) 

c 1 

 

s = 0 

5 

From equation(2.13), the dimension 

( p q v cs 

g is 

 

5 

)! ( q 

( q 1)! ( 1) 

u )! ( u v cs 

q 

5 

)! ( q 

 

p 

 

s 

5 

)! s 

5 

.......... ...(3.2.1) 

! 

Dim g 

 

= 

 

d | 

1 

d 

( d ) W 

( J ) 

 

 

 

 

d 

 

 

 



= 

 

d | 

1 

d 

( d ) 

 

( J ) 

nT 

 

d 

(| n | 1)! 

n! 

 

D ( i ) 

n 

i 

, 

( J ) 


Dim 

g 

= 

 

d | 

1 

d 

( d ) 

v 

min ( p / d , q / d , u / d ,[ ]) 

d ( c 1) 

 

s = 0 

5 

( u v s ) 0 

5 

p 

( 

d 

 

q 

d 

 

v 

d 

c 

s 

d 

5 

)! ( 

q 

d 

q 

( 

d 

 

u 

d 

1)! ( 1) 

)! ( 

u 

d 

q / d 

 

v 

c 

d 

s 

d 

5 

)! ( 

q 

d 

 

p 

d 

 

s s 

5 5 

)! ! 

d d 

) 

If we solve the same P ( H 

( J 

) using partition and substituting the partition, we have 

T 

( J ) 

( ) = { p q , q u , q p , u , }, 

1 

2 

1 2 

where is partition of v with parts upto (1, c 1) and of length v and is the partition of s with parts 

1 

2 

5 

of s of length s . 

5 

5 

( J ) 

Applying T ( ) in Witt partition formula (eqn.(2.12)) 

W 

( J ) 

( ) = 

 

( J ) 

T 

( ) 

From equation(2.13), the dimension 

( q 1)! ( 1) 

( p q )! ( q u )! ( q 

Dim 

g 

1 

g is 

 

 

= 

= 

 

d | 

 

d | 

1 

( d ) W 

d 

1 

( d ) 

d 

( J ) 

 

 

 

 

q 

p 

 

d 

( J ) 

nT 

 

d 

 

 

 

2 

)! ( 

(| n | 1)! 

n! 

u ) | 

1 

 

2 

D ( i ) 

.......... 

|! 

( J ) 


n 

i 

, 

........(3 

.2.2) 

Dim 

g 

 

= 

1 

( q / d 1)! ( 1) 

( d ) 

( ) ( / / )! ( / / )! ( / / )! ( / ) | 

| d 

J p d q d q d u d q d p d u d 

 

d 

, T 

1 2 

( ) 

1 

q / d 

2 

1 

2 

|! 

Now consider the equation 

v 

min ( p , q , u ,[ ]) 

c 1 

 

s = 0 

5 

( p 

q 

v cs 

5 

)! ( q 

( q 1)! ( 1) 

u )! ( u v cs 

q 

5 

)! ( q 

 

p 

 

s 

5 

)! s 

5 

. 

! 

= 

( p 

 

q 

 

( q 1)! ( 1) 

v )! ( q u )! ( u v )! ( q 

q 

 

p )! 

 

( p 

 

q 

 

v 

( q 1)! ( 1) 

c )! ( q u )! ( u v c )! ( q 

q 

 

p 

1)!1! 

 

( p 

 

q 

 

v 

( q 1)! ( 1) 

2 c )! ( q u )! ( u v 

q 

2 c )! ( q 

 

p 

 

2)!2! 

 

( p 

 

q 

 

( q 1)! ( 1) 

v 3 c )! ( q u )! ( u v 3 c )! ( q 

q 

 

p 

3)!3! 

....... upto s 

5 

= 

min 

v 

( p , q , u , [ 

c 

]). 

1 



= 

 

( J ) 

, T 

( ) 

1 2 

( q 1)! ( 1) 

( p q )! ( q u )! ( q 

1 

q 

p )! ( u ) | |! 

where is partition of v with parts upto (1, c 1) and of length v and is the partition of s with parts 

1 

2 

5 

of s of length s . 

5 

5 

Hence we proved the following theorem. 

Theorem 3.2.1: For the extended-hyperbolic Borcherds superalgebra g = g ( A , m , C ) associated with the 

extended-hyperbolic Borcherds-Cartan super matrix 

= = p q u v Q . Then the dimension of 

1 

2 

3 

4 

 

Dim 

g 

= 

 

d | 

1 

( d ) 

d 

v 

min ( p / d , q / d , u / d ,[ ]) 

d ( c 1) 

 

s = 0 

5 

( u v s ) 0 

5 

p 

( 

d 

q 

d 

2 

1 

2 

k a b c 

 

 

a 2 1 0 

A = 

let us consider 

 

b 1 2 1 

 

 

 

c 0 2 2 

g is 

q 

( 

d 

v s q 

5 

c )! ( 

d d d 

u 

d 

1)! ( 1) 

)! ( 

u 

d 

q / d 

v s q 

5 

c )! ( 

d d d 

p 

d 

. 

s s 

5 5 

)! ! 

d d 

Moreover the following combinatorial identity holds: 

v 

min ( p , q , u ,[ ]) 

c 1 

 

s = 0 

5 

( p 

 

q 

 

v 

cs 

5 

)! ( 

q 

( q 1)! ( 1) 

u )! ( u v cs 

q 

5 

)! ( 

q 

 

p 

 

s 

5 

)! s 

5 

= 

! 

 

( J ) 

, T 

( ) 

1 2 

( q 1)! ( 1) 

( p q )! ( q u )! ( q 

1 

q 

p 

2 

)! ( 

u ) | 

1 

2 

.......... 

|! 

.(3 .2 .3) 

where is partition of v with parts upto (1, c ) and of length v and is the partition of s with parts of 

1 

2 

5 

s of length s . 

5 

5 

k 1 b c 

 

 

1 2 1 0 

Example 3.2.2: For the Borcherds-Cartan super matrix A = 

, 

 

b 1 2 1 

 

 

 

c 0 2 2 

consider the root = = (7,6,4,3) = ( p , q , u , v ) with c=1. Applying in equation (3.2.1), we have 

v 

min ( p , q , u ,[ ]) 

c 1 

= 

 

s = 0 

5 

1 

 

s = 0 

5 

( p q v cs 

(7 6 3 s 

= 

5! (1) 

 

4!2!1!0! 

5 

5! (1) 

5!2!0!0!1! 

5 

( q 1)! ( 1) 

)! ( q u )! ( u v cs 

(6 1)! ( 1) 

)! (6 4)! (4 3 s 

)! ( q 


6 

5 

q 

)! (6 7 s 

5 

5 

)! s 

p s 

= 

5 

 

1 

= 3. 

2 2 

Consider the root = as (7,6,4,3) with c=1. Applying in equation (3.2.2), we have 

5 

! 

5 

)! s 

5 

!


 

( J ) 

, T 

( ) 

1 2 

( p 

 

q 

)! ( q 

1 

( q 1)! ( 1) 

u )! ( q 

q 

p 

2 

)! ( 

u 

) | 

1 

2 

|! 

= 

(7 

(6 

6 3)! (6 

1)! ( 1) 

4)! (4 3)! (6 

6 

7)!0! 

 

(7 

(6 

6 3 1)! (6 

1)! ( 1) 

4)! (4 3 1)! (6 

6 

7)!1! 

= 

5 

 

1 

= 3. 

2 2 

Hence the equality (3.2.3) holds. 

Remark: In the above two sections 3.1 and 3.2., the identities (3.1.4) and (3.2.3) hold for any root, because we 

( J ) 

have derived the identities by simply solving the T ( ) in two different ways. 

Remark: It is hoped that, in general, superdimensions of roots and the corresponding combinatorial identities 

for Borcherds Superalgebras which are extensions of all finite dimensional Kac-Moody algebras and 

superdimensions for all other categories can also be found out. 

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(2) 

2

Combinatorial Identities Related To Root Supermultiplicities In Some Borcherds Superalgebras

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