combinatorial identities related to root supermultiplicities in some borcherds superalgebras
DESCRIPTION
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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
www.theijes.com The IJES Page 63
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 2
B and 3
B are found out.
Keywords: Borcherds superalgebras, Colored superalgebras, Root supermultiplicities.
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Date of Submission: 10-June-2015 Date of Accepted: 25-June-2015
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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 2
A and 3
A were considered in Sthanumoorthy et
al.(2009a) and therein dimension formulas were found out.In Sthanumoorthy and Priyadharsini(2014), we have
computed combinatorial identities for 2
A and 3
A and root super multiplicities for some hyperbolic Borcherds
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 2
B and 3
B .
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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 Ijiij
aA,
)(= is called
Borcherds-Cartan matrix if it satisfies:
(1) 2=ii
a or 0ii
a for all Ii ,
(2) 0ij
a if ji and Zij
a if 2=ii
a ,
(3) 0=0=jiij
aa
We say that an index i is real if 2=ii
a and imaginary if 0ii
a . We denote by
0}.|{=2},=|{=ir
ii
m
ii
eaIiIaIiI
Let }|{=0>
Iimmi
Z be a collection of positive integer such that 1=i
m for all e
Iir
. 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= IiiagDi
with )(0> Iii
such that DA is symmetric.
Let Ijiij
cC,
)(= be a complex matrix satisfying 1=jiij
cc for all Iji , . Therefore, we have 1= ii
c
for all Ii . We call Ii an even index if 1=ii
c and an odd index if 1= ii
c .
We denote by ven
Ie
( )o dd
I the set of all even (odd) indices.
Definition 2.3: A Borcherds-cartan matrix Ijiij
aA,
)(= is restricted (or colored) with respect to C if it
satisfies:
If 2=ii
a and 1= ii
c then ij
a are even integers for all Ij . In this case, the matrix C is called a
coloring matrix of A .
Let )()(=iIiiIi
dh CCh
be a complex vector space with a basis };,{ Iidhii
, and for each
Ii , define a linear functional *hi
by
)1.2......(...........=)(,=)( Ijallfordahijjijiji
If A is symmetrizable, then there exists a symmetric bilinear form (|) on *h satisfying
jijijijiaa ==)|( for all Iji , .
Definition 2.4: Let iIi
Q Z
= and iIi
Q 0
= Z ,
QQ = . 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 Ijiij
cC,
)(= defines a bimultiplicative form X
CQQ : by
,,f=),( Ijalliorcijji
),,(),(=),(
),(),(=),(
for all Q ,, . Note that satisfies
)2.2....(..........,,f1=),(),( Qorall
since 1=jiij
cc for all Iji , . In particular 1=),( for all Q .
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
LLQ
= together with a
bilinear product LLL :][, satisfying
,,
LLL
],,)[,(=, xyyx
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]],[,)[,(]],,[[=,, zxyzyxzyx
for all Q , and LzLyLx ,,
.
In a -colored Lie super algebra
LLQ
= , for
Lx , we have 0=],[ xx if is even and
0=]],[,[ xxx if is odd.
Definition 2.6: The universal enveloping algebra )( LU of a -colored Lie super algebra L is defined to be
JLT )/( , where )( LT is the tensor algebra of L and J is the ideal of )( LT generated by the elements
xyyxyx ),(],[ ),(
LyLx .
Definition 2.7: The Borcherds superalgebra ),,(= CmAgg associated with the symmetrizable Borcherds-
Cartan matrix A of charge );(= Iimmi
and the coloring matrix Ijiij
cC,
)(= is the -colored Lie
super algebra generated by the elements ),1,2,=,(,),(,iikikii
mkIifeIidh with defining
relations:
0=0=,=,
,2=0=)(=)(
=,
,=,,=,
,=,,=,
0,=,=,=,
11
ijjlikjlik
iijl
ija
ikjl
ija
ik
iklijjlik
jlijjlijlijjli
jlijjlijlijjli
jijiji
aifffee
jiandaiffadfeade
hfe
ffdeed
fafheaeh
dddhhh
for ji
mlmkIji ,1,=,,1,=,, .
The abelian subalgebra )()(=iIiiIi
dh CCh
is called the Cartan subalgebra of g and the linear
functionals )( Iii
*h defined by (2.1) are called the simple roots of g . For each e
Iir
, let
)(h*GLri be the simple reflection of *h defined by
).()(=)( h* iii
hr
The subgroup W of )(h*GL generated by the i
r ’s )(re
Ii is called the Weyl group of the Borcherds super
algebra g .
The Borcherds superalgebra ),,(= CmAgg has the root space decomposition
ggQ
= , where
}.)(=],[|{= hgg hallforxhxhx
Note that i
miii
ee,,1
= CCg
and i
miii
ff,,1
= CCg
We say that X
Q is a root of g if 0
g . The subspace
g is called the root space of g attached to
. A root is called real if 0>)|( and imaginary if 0)|( .
In particular, a simple root i
is real if 2=ii
a that is if e
Iir
and imaginary if 0ii
a that is if m
Iii
.
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
, and
the set
of all roots, positive roots and negative roots, respectively. Also we denote 0
(1
) the set of all even (odd)
roots of g . Define the subspaces
gg
= .
Then we have the triangular decomposition of g : .=
ghgg
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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
,=
VV h
where .)(=|= h hallvforhvhVvV
If 0,
V then is called a weight of V , and
Vdim 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 Vv
such that
1. 0,=
veik for all Ii ,
imk ,1,= ,
2.
vhvh )(= for all hh and
3.
vUV )(= g .
The vector
v is called a highest weight vector.
For a highest weight module V with highest weight , we have
(i)
vUV
)(= g ,
(ii)
vVVV C=,=
and
(iii) <dim
V for all .
Definition 2.11: Let )(VP denote the set of all weights of V . When all the weights spaces are finite
dimensional, the character of V is defined to be ,)dim(=c
eVhV
*h
where
e are the basis elements of the group ][h*C with the multiplication given by
eee = for
*h , . Let
gh=b be the Borel subalgebra of g and
C be the 1-dimensional
b -module
defined by )1(=10,=1 hh
g for all .hh The induced module
Cg)(
)(=)(
bU
UM 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 )()/(=)( JMV is irreducible.
Let
P be the set of all linear functionals *h satisfying
m
i
dde
i
e
i
Iiallorh
IIiallorh
Iiallorh
i
or
0
r
0
f0)(
f2)(
f)(
Z
Z
The elements of
P are called the dominant integral weights.
Let *h be the C -linear functional satisfying iii
ah2
1=)( for all Ii . Let T denote the set of all
imaginary simple roots counted with multiplicities, and for ,TF we write ,F if 0=)(i
h for all
.Fi
Definition 2.12:[Kang and Kim (1997)] Let J be a finite subset of e
Ir
. We denote by
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JJjJjJ
=),(= Z and .\=)(
J
J Let
(2.3),=)(
0
ghg
J
J
and
.=
)(
)(
gg
J
J
Then )(
0
Jg is the restricted Kac-Moody super algebra (with an extended Cartan subalgebra) associated with the
Cartan matrix JjiijJ
aA,
)(= and the set of odd indices dddd
IJJoo
=
1}=|{= jj
cJj
Then the triangular decomposition of g is given by .=)()(
0
)( JJJ
gggg
Let JjrWjJ
|= be the subgroup of W generated by the simple reflections ),( Jjrj
and let
)(|=)( JWwJWw
where 0}.(2.4)<|{=1
w
w
Therefore J
W is the Weyl group of the restricted Kac-Moody super algebra )(
0
Jg and )( JW is the set of right
coset representatives of J
W in .W That is ).(= JWWWJ
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).
Lemma 2.13: Suppose j
rww = and 1.)(=)( wlwl Then )( JWw if and only if )( JWw and
).()( JwJ
Let 0,1)=(=,
iJiJi
and 0,1).=(\=)(,
iJJiii
Here )(
10
denotes the set of all
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 e
Ir
. Then
)))(((c1)(=
)(1
)(1
||)(
)(
dim
)(1
dim
)(0
FswhV
e
e
J
Fwl
TF
JWw
J
J
g
g
where )( J
V denotes the irreducible highest weight module over the restricted Kac-Moody super algebra
)(
0
Jg with highest weight and where F runs over all the finite subsets of T such that any two elements of
F are mutually perpendicular. Here )( wl denotes the length of ||, Fw the number of elements in F , and
)( Fs the sum of the elements in F .
Definition 2.15: A basis elements of the group algebra ][å
hC by defining .),(=
eE
Also define the super dimension
gDim of the root space
g by
(2.5)...........................dim),(=
ggDim
Since ))(( Fsw is an element of
Q , all the weights of the irreducible highest weight )(
0
Jg -module
)))((( FswVJ
are also elements of
Q .
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Hence one can define the super dimension
DimV of the weight space
V of )))((( FswVJ
in a
similar way. More generally, for an h -diagonalizable )(
0
Jg -module
VVå
h
= such that QVP )( , we
define the super dimension
DimV of the weight space
V to be
(2.6)dim),(=
VDimV
For each 1k , let
(2.7)..........................)))(((=
|=|)(
)(
)(
FswVHJ
kFwl
TF
JWw
J
k
and define the homology space )( J
H of )( J
g to be
(2.8),=1)(=)(
3
)(
2
)(
1
)(1
1=
)(
JJJJ
k
k
k
JHHHHH
an alternating direct sum of the vector spaces.
For
Q , define the super dimension )( J
DimH
of the -weight space of )( J
H to be
)(1)(=)(1
1=
)( J
k
k
k
JDimHDimH
)))(((1)(=
|=|)(
)(
1
1=
FswDimVJ
kFwl
TF
JWw
k
k
2.9)........(..........)))(((1)(=1||)(
1||)(
)(
FswDimVJ
Fwl
Fwl
TF
JWw
Let
(2.10)...................0dim|)(=)()()(
JJHJQHP
and let ,,,,321 be an enumeration of the set )(
)( JHP . Let
)(=)(
J
i
DimHiD
.
Remark: The elements of )()( J
HP can be determined by applying the following proposition, proved in Kac
(1990).
Proposition 2.16: (Kang and Kim, 1997)
Let .
P Then }•|.{=)(
torespectwithatenondegenerisPWP .
Now for ),( JQ
one can define
(2.11).......................,=,|)(==)(01
)(
iiiii
JnnnnT
Z
which is the set of all partitions of into a sum of i
’s. For ),()(
JTn use the notations
inn |=| and
!=!i
nn .
Now, for ),( JQ
the Witt partition function )()(
JW is defined as
(2.12).......................)(!
1)!|(|=)(
)()(
)( in
JTn
JiD
n
nW
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Now a closed form formula for the super dimension
gDim of the root space ))(( J
g is given by
the following theorem. The proof is given in Kang and Kim(1997).
Theorem 2.17: Let J be a finite subset of e
Ir
. Then, for )( J
, we have
dWd
dDim
J
d
)(
|
)(1
=g
.(2.13)..........)(!
1)!|(|)(
1=
)(|
in
d
JTn
d
iDn
nd
d
where is the classical Möbius function. Namely, for a natural number ,n )( n is defined as follows:
freesquarenotisitf
primesistinctppppnor
nor
nkk
k
i0
),d:,,(=f1)(
1,=f1
=)(11
and, for a positive integer |, dd 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 2
B and 3
B (with multiplicity 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 2
B
Below, we find the dimension formulae and combinatorial identities for the Borcherds superalgebras which are
extension of .2
B (for a same set re
J ) by solving )()(
JT in two different method.
Consider the extended-hyperbolic Borcherds superalgebra ),,(= CmAgg associated with the extended-
hyperbolic Borcherds-Cartan super matrix,
22
12=
b
a
bak
A and the corresponding coloring matrix C=
1
1
1
1
3
1
2
3
1
1
21
cc
cc
cc
with .,,321
XCccc
Let {1,2,3}=I be the index set for the simple roots of .g Here 1
is the imaginary odd simple root with
multiplicity 1 and 32
, are the real even simple roots.
Let us consider the root .=332211
Qkkk We have .1)(=),(2
1k
Hence is an even
root(resp. odd root) if 1
k is even integer(resp. odd integer). Also }{=1
T and the subset TF is either
empty or }.{1
Take re
J as {3}.=J By Lemma 2.13, this implies that }.{1,=)(2
rJW From the
equations(2.7) and (2.8), the homology space can be written as
)()(=21
)(
1
JJ
JVVH
)1)((=21
)(
2 aVH
J
J
30=)(
kHJ
k
)(
2
)(
1
)(=
JJJHHHTherefore
)1)(()()(=2121
aVVVJJJ
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with 1=1;=)(
(0,1,0)
)(
(1,0,0)
JJDimHDimH
.1=1;=)(
1)(1,1,
)(
(1,1,1)
J
a
JDimHDimH
We take 1)}.(1,1,(1,1,1),(0,1,0),{(1,0,0),=)()(
aHPJ
Let ,==321 Quqp with .),,(
000 ZZZtqp Then by proposition.(2.16)
we get )}.,,(=1)(1,1,(1,1,1)(0,1,0)(1,0,0)|),,,{(=)(43214321
)(uqpassssssssT
J
This implies psss =431
qsss =432
usas =1)(43
We have 41
= asups
42
= asuqs
43
1)(= saus
])1
[,,(0=4
a
uqpmintos
Applying 4321
,,, ssss in Witt partition formula(eqn.(2.12)), we have,
...(3.1.1)..........!)!1)(()!()!(
1)(1)!(=)(
444
4
4
])1
[,,(
0=4
)(
ssauasuqqp
asqupW
asqupa
uqpmin
s
J
From eqn(2.13), the dimension of
g is
dWd
dDim
J
d
)(
|
)(1
=g
,)(!
1)!|(|)(
1=
)(|
in
d
JTn
d
iDn
nd
d
Substituting the value of )()(
JW from eqn. (3.1.1) in the above dimension formula, we have,
)2.1.3..(...........
)!/()!1)(
()!()!(
1)(1)!(
)(1
=
444
/4
///
4
])1)(
[,,(
0=4
|dss
d
a
d
us
d
a
d
u
d
q
d
q
d
p
sd
a
d
q
d
u
d
p
dd
Dim
dasdqdudp
ad
t
d
q
d
pmin
sd
g
If we solve the same ),()( J
HP using partition and substituting this partition in )()(
JT , we have
},,1)(,),{(=)(221
)( auuqpT
J
where 1
is the partition of u ` with parts 1)(1, a of length ‘u’ and 2
is the partition of 4
s with parts upto
min ])1
[,,(a
uqp . Applying )(
)(
JT in Witt partition formula(eqn.(2.12)), we have
.1.3)........(3..........!)!1)(()!()!(
1)(1)!(=)(
221
1
1
)()(
2,
1
)(
auuqp
upW
up
JT
J
From eqn.(2.13), the dimension of
g is
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dWd
dDim
J
d
)(
|
)(1
=g
,)(!
1)!|(|)(
1=
)(|
in
d
JTn
d
iDn
nd
d
Substituting the value of )()(
JW from eqn. (3.1.2) in the above dimension formula, we have
)!/())!/1)((()!()!(
1)(1)!(
)(1
=
22
1
/1
//1
)()(
2,
1)(|
ddad
u
d
u
dd
q
d
p
dd
u
d
p
dd
Dim
ddudp
JT
d
JTn
d
g
Which gives another form of dimension of
g.
Applying the value of s4 in (3.1.1), we get
!)!1)(()!()!(
1)(1)!(=)(
444
4
4
])1
[,,(
0=4
)(
ssauasuqqp
asqupW
asqupa
uqpmin
s
J
)!()!()!(
1)(1)!(
uuqqp
qupqup
])1/[,,min(.......!1))!1(()!()!(
1)(1)!(
auqpuptoauauqqp
aqupaqup
.!)!1)(()!()!(
1)(1)!(=
221
1
1
)()(
2,
1
auuqp
upup
JT
where 1
is the partition of u ` with parts )(1, a of length ‘4
asu ’ and 2
is the partition of 4
s with parts
upto min ])1
[,,(a
uqp .
Hence, we get the following theorem.
Theorem 3.1.1: For the extended-hyperbolic Borcherds superalgebra ),,(= CmAgg associated with the
extended-hyperbolic Borcherds-Cartan super matrix
22
12=
b
a
bak
A with charge {1,1,1},=m
consider the root
.=321 Quqp Then the dimension of
g is
.
)!/()!1)(
()!()!(
1)(1)!(
)(1
=
444
/4
///
4
])1)(
[,,(
0=4
|dss
d
a
d
us
d
a
d
u
d
q
d
q
d
p
sd
a
d
q
d
u
d
p
dd
Dim
dasdqdudp
ad
t
d
q
d
pmin
sd
g
Moreover the following combinatorial identity holds:
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4).....(3.1.!)!1)(()!()!(
1)(1)!(=
!)!1)(()!()!(
1)(1)!(
221
1
1
)()(
2,
1444
4
4
])1
[,,(
0=4
auuqp
up
ssauasuqqp
asqupup
JT
asqupa
uqpmin
s
where 1
is the partition of u ` with parts )(1, a of length ‘4
asu ’ and 2
is the partition of 4
s with parts
upto min ])1
[,,(a
uqp .
Example 3.1.2: For the Borcherds-Cartan super matrix ,
22
121
1
=
b
bk
A consider a root
(5,4,3)== with a=1.
Substituting 1=(5,4,3),== a in eqn(3.1.1), we have
!)!1)(()!()!(
1)(1)!(=)(
444
4
4
])1
[,,(
0=4
)(
ssauasuqqp
asqupW
asqupa
uqpmin
s
J
!)!2(3)!3(44)!(5
1)(1)!43(5=
444
4435
4
])2
3[(5,4,
0=4
sss
ssmin
s
2500.=3.4.5.6.74.5=1!2!1!1!
1)(7!
1!1!3!
1)(5!=
76
Substituting 1=(5,4,3),== a in eqn(3.1.3), we have
!)!1)(()!()!(
1)(1)!(=)(
221
1
1
)()(
2,
1
)(
auuqp
upW
up
JT
J
2500.=1!2!1!1!
1)(7!
1!1!3!
1)(5!=
76
Hence the equality (3.1.4) holds.
3.2.Dimension Formula and combinatorial identity for the Borcherds superalgebra which is an extension
of 3
B
Here we are finding the superdimension Formula and combinatorial identity for the Borcherds superalgebra
which is an extension of 3
B using the same re
J and solving )()(
JT in two different ways.
Consider the extended-hyperbolic Borcherds superalgebra ),,(= CmAgg associated with the extended-
hyperbolic Borcherds-Cartan super matrix
220
121
012=
c
b
a
cbak
A and the corresponding coloring
matrix C=
1
1
1
1
1
6
1
5
1
3
6
1
4
1
2
54
1
1
321
ccc
ccc
ccc
ccc
with .,,,4321
XCcccc
Combinatorial Identities Related To Root…
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Let {1,2,3,4}=I be the index set with charge {1,1,1,1}.=m
Let us consider the root .=44332211
Qkkkk We have .1)(=),(2
1k
Hence is
an even root(resp. odd root) if 1
k is even integer(resp. odd integer). Also }{=1
T and the subset TF is
either empty or }.{1
Take re
J as {2,3}.=J By Lemma 2.13, this implies that }.{1,=)(4
rJW
From the equations (2.7) and (2.8),the homological space can be written as
))(())(1(=41
)(
1 rVVH
JJ
J
)()(=41
JJ
VV
)1)((=)(412
cVJHJ
30=)(
kHJ
k
)1)(()()(==4141
)(
1
)( cVVVHThereforeH
JJJ
JJ!
with
1;=1;=1;=)(
(1,1,1,0)
)(
(1,1,0,0)
)(
(1,0,0,0)
JJJDimHDimHDimH
1=1;=)(
1)(1,1,1,
)(
(1,1,1,1)
J
c
JDimHDimH
Hence we have
1)}.(1,1,1,(1,1,1,1),(1,1,1,0),(1,1,0,0),,{(1,0,0,0)=)()(
cHPJ
Let ,==4321 Qvuqp with .),,,(
0000 ZZZZvuqp Then by
proposition.(2.16), we get
(1,1,1,0)(1,1,0,0)(1,0,0,0)|),,,,{(=)(32154321
)(ssssssssT
J
)}.,,,(=1)(1,1,1,(1,1,1,1)54
vuqpcss
This implies pssss =4321
qssss =5432
usss =543
.=1)(54
vscs
We have 51
= csvqps
uqs =2
53
= csvus
54
= spqs
]).1
[,,,(0=5
c
vuqpmintos
Applying 54321
,,,, sssss in Witt partition formula(eqn.2.12), we have
...(3.2.1)..........!)!()!()!()!(
1)(1)!(=)(
5555
])1
[,,,(
0=5
)(
sspqcsvuuqcsvqp
qW
qc
vuqpmin
s
J
From equation(2.13), the dimension
g is
dWd
dDim
J
d
)(
|
)(1
=g
Combinatorial Identities Related To Root…
www.theijes.com The IJES Page 74
,)(!
1)!|(|)(
1=
)(|
in
d
JTn
d
iDn
nd
d
Substituting the value of )()(
JW from eqn. (3.3.1) in the above dimension formula, we have,
!)!()!()!()!(
1)(1)!(
)(1
=
5555
/])1)(
[,/,/,/(
0)5
(
0=5
|
d
s
d
s
d
p
d
q
d
sc
d
v
d
u
d
u
d
q
d
sc
d
v
d
q
d
p
d
q
dd
Dim
dq
cd
vdudqdpmin
svu
sd
g
If we solve the same )()( J
HP using partition and substituting the partition, we have
},,,,,{=)(2121
)( upquqqpT
J
where 1
is partition of v with parts upto 1)(1, c and of length v and 2
is the partition of 5
s with parts
of 5
s of length .5
s
Applying )()(
JT in Witt partition formula (eqn.(2.12))
.2.2)........(3..........|!|)()!()!()!(
1)(1)!(=)(
2121)()(
)(
upquqqp
qW
q
JT
J
From equation(2.13), the dimension
g is
dWd
dDim
J
d
)(
|
)(1
=g
,)(!
1)!|(|)(
1=
)(|
in
d
JTn
d
iDn
nd
d
Substituting the value of )()(
JW from eqn. (3.2.2) in the above dimension formula, we have,
|!|)/()!//()!//()!//(
1)(1)!/()(
1=
2121
/
)()(
2,
1|
dudpdqdudqdqdp
dqd
dDim
dq
JT
d
g
Now consider the equation
.!)!()!()!()!(
1)(1)!(
5555
])1
[,,,(
0=5
sspqcsvuuqcsvqp
qqc
vuqpmin
s
)!()!()!()!(
1)(1)!(=
pqvuuqvqp
1)!1!()!()!()!(
1)(1)!(
pqcvuuqcvqp
2)!2!()!2()!()!2(
1)(1)!(
pqcvuuqcvqp
3)!3!()!3()!()!3(
1)(1)!(
pqcvuuqcvqp
]).1
[,,,(=.......5
c
vuqpminsupto
Combinatorial Identities Related To Root…
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|!|)()!()!()!(
1)(1)!(=
2121)()(
2,
1
upquqqp
JT
where 1
is partition of v with parts upto 1)(1, c and of length v and 2
is the partition of 5
s with parts
of 5
s of length .5
s
Hence we proved the following theorem.
Theorem 3.2.1: For the extended-hyperbolic Borcherds superalgebra ),,(= CmAgg associated with the
extended-hyperbolic Borcherds-Cartan super matrix
220
121
012=
c
b
a
cbak
A let us consider
Qvuqp
4321== . Then the dimension of
g is
.
!)!()!()!()!(
1)(1)!(
)(1
=
5555
/])1)(
[,/,/,/(
0)5
(
0=5
|
d
s
d
s
d
p
d
q
d
sc
d
v
d
u
d
u
d
q
d
sc
d
v
d
q
d
p
d
q
dd
Dim
dq
cd
vdudqdpmin
svu
sd
g
Moreover the following combinatorial identity holds:
)3.2.3.(..........|!|)()!()!()!(
1)(1)!(
=!)!()!()!()!(
1)(1)!(
2121)()(
2,
1
5555
])1
[,,,(
0=5
upquqqp
q
sspqcsvuuqcsvqp
q
q
JT
qc
vuqpmin
s
where 1
is partition of v with parts upto )(1, c and of length v and 2
is the partition of 5
s with parts of
5s of length .
5s
Example 3.2.2: For the Borcherds-Cartan super matrix ,
220
121
0121
1
=
c
b
cbk
A
consider the root = = ),,,(=(7,6,4,3) vuqp with c=1. Applying in equation (3.2.1), we have
!)!()!()!()!(
1)(1)!(
5555
])1
[,,,(
0=5
sspqcsvuuqcsvqp
qqc
vuqpmin
s
!)!7(6)!3(44)!(6)!36(7
1)(1)!(6=
5555
61
0=5
sssss
5!2!0!0!1!
(1)5!
4!2!1!0!
(1)5!=
3.=2
1
2
5=
Consider the root = as (7,6,4,3) with c=1. Applying in equation (3.2.2), we have
Combinatorial Identities Related To Root…
www.theijes.com The IJES Page 76
7)!1!(61)!3(44)!(61)!36(7
1)(1)!(6
7)!0!(63)!(44)!(63)!6(7
1)(1)!(6=
|!|)()!()!()!(
1)(1)!(
66
2121)()(
2,
1
upquqqp
JT
3.=2
1
2
5=
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
have derived the identities by simply solving the )()(
JT 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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