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FREE LIE ALGEBRAS, GENERALIZED WITT

FORMULA, AND THE DENOMINATOR IDENTITY

Seok-Jin Kang and Myung-Hwan Kim

Department of MathematicsSeoul National University

Seoul 151-742, Korea

Abstract. Let Γ be a countable abelian semigroup satisfying a suitable finitenesscondition, and let L =

Lα∈Γ Lα be the free Lie algebra generated by a Γ-graded

vector space V over C. In this paper, from the denominator identity, we derivea dimension formula for the homogeneous subspaces of the free Lie algebra L =L

α∈Γ Lα, and discuss numerous applications of our dimension formula to variousinteresting cases. Our dimension formula will be expressed in terms of the Wittpartition functions. Various expressions of the Witt partition functions will give riseto a number of interesting combinatorial identities. As a special case, we obtain arecursive relation for the coefficients of the elliptic modular function j.

Introduction

Let V be an r-dimensional vector space over C and let L = F(V ) be the free Liealgebra generated by V . Then the Lie algebra L has a Z>0-gradation L =

⊕∞n=1 Ln

by setting degv = 1 for all v ∈ V . The dimensions of the homogeneous subspacesLn are given by the Witt formula

dimLn =1n

∑

d|n

µ(d)rn/d,

where µ is the classical Mobius function (cf. [Bo], [J], [Se]). The main ingredientof the proof of the Witt formula is the following identity:

∞∏

n=1

(1− qn)dimLn = 1− rq,

This research was supported in part by Basic Science Research Institute Program, Ministry ofEducation, BSRI-95-1414 and POSCO Research Fund at Seoul National University, Korea.1991 Mathematics Subject Classifications: 17B01, 17B65, 17B70, 11F22.

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1

2 SEOK-JIN KANG AND MYUNG-HWAN KIM

which is a consequence of the Poincare-Birkhoff-Witt Theorem. We will inter-pret the above identity as the denominator identity for the free Lie algebra L =⊕∞

n=1 Ln.

In this paper, we work in a much more general setting. Suppose Γ is a countableabelian semigroup such that every element α ∈ Γ can be written as a sum ofelements in Γ only in finitely many ways. Let V =

⊕

α∈Γ Vα be a Γ-graded vectorspace over C with dimVα = d(α) < ∞ for all α ∈ Γ, and let L = F(V ) bethe free Lie algebra generated by V . Then the Lie algebra L has a Γ-gradationL =

⊕

α∈Γ Lα induced by that of V . The main purpose of this paper is to derivea closed form dimension formula for the homogeneous subspaces Lα (α ∈ Γ), andto discuss a number of applications of our dimension formula to various interestingcases.

In Section 1, we start with an interpretation of the Euler-Poincare principle forthe homology modules over graded Lie algebras as the denominator identity for thegraded Lie algebras. In particular, if L =

⊕

α∈Γ Lα is the free Lie algebra generatedby a Γ-graded vector space V =

⊕

α∈Γ Vα over C with dimVα = d(α), then thedenominator identity for the free Lie algebra L is equal to

∏

α∈Γ

(1− eα)dimLα = 1− chV = 1−∑

α∈Γ

d(α)eα.

Now, using the formal power series log(1−t) = −∑∞

k=1tk

k and the Mobius inversion,we will derive from the denominator identity a nice closed form dimension formulafor the homogeneous subspaces Lα (α ∈ Γ) of L (Theorem 1.1. See also [Ka1]and [Ju]). Our dimension formula will be expressed in terms of the Witt partitionfunctions associated with the partitions of the elements in Γ. In many cases, bytaking the logarithm of the righthand side of the denominator identity, we will beable to obtain simple expressions of the Witt partition functions. Moreover, variousexpressions of the Witt partition functions will give rise to a number of interestingcombinatorial identities.

In Section 2, we recover the Witt formula as a special case of our dimensionformula, and consider a “1-dimensional generalization” of the Witt formula. Moreprecisely, let V =

⊕∞i=1 Vi be a Z>0-graded vector space over C with dimVi =

d(i) < ∞ for all i ≥ 1, and let L =⊕∞

n=1 Ln be the free Lie algebra generated byV . Then the dimensions of the homogeneous subspaces Ln are given by the formula

dimLn =∑

d|n

1dµ(d)

∑

s∈T (n/d)

(|s| − 1)!s!

∏

d(i)si ,

where T (n) denotes the set of all partitions of n into a sum of positive integers(Theorem 2.2).

FREE LIE ALGEBRAS 3

Similarly, in Section 3, we consider a “2-dimensional generalization” of the Wittformula. That is, let V =

⊕∞i,j=1 V(i,j) be a Z>0×Z>0-graded vector space over C

with dimV(i,j) = d(i, j) < ∞ for all i, j ≥ 1, and let L =⊕∞

m,n=1 L(m,n) be the freeLie algebra generated by V . Then the dimensions of the homogeneous subspacesL(m,n) are given by the formula

dimL(m,n) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

∏

d(i, j)sij ,

where T (m,n) denotes the set of all partitions of (m,n) into a sum of ordered pairsof positive integers (Theorem 3.1). Applying the above generalizations of the Wittformula to various special cases, we obtain simple expressions of the Witt partitionfunctions. Moreover, we derive a number of combinatorial identities which relatethe various Witt partition functions.

Finally, in Section 4, we interpret the product formulas for formal power series asthe denominator identities for graded Lie algebras. As a special case, we considerthe elliptic modular function

j(q)− 744 =∞∑

n=−1

c(n)qn = q−1 + 196884q + 21493769q2 + · · · .

As a consequence of the product formula for j(q)− 744 obtained in [B2], we derivea product formula

∞∏

m,n=1

(1− pmqn)c(mn) = 1−∞∑

i,j=1

c(i + j − 1)piqj .

Now, let V =⊕∞

i,j=1 V(i,j) be a Z>0 × Z>0-graded vector space over C withdimV(i,j) = c(i + j − 1) for all i, j ≥ 1, and let L =

⊕∞m,n=1 L(m,n) be the free

Lie algebra generated by V . Then the above product formula can be interpretedas the denominator identity for the free Lie algebra L. Applying our dimensionformula to this setting, we obtain a complete recursive relation for the coefficientsc(n) of the elliptic modular function j(q)− 744 (Corollary 4.2):

c(mn) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

∏

c(i + j − 1)sij .

The above relation was obtained independently by Kang [Ka2] and Jurisich [Ju]using the theory of generalized Kac-Moody algebras. Let M be the Monster simplegroup and let V \ =

⊕∞n=−1 V \

n be the Moonshine Module for the Monster simple


group M ([FLM]). Then, in [JLW] and [KK], the the above relation was generalizedto the relations of the coefficients cg(n) (g ∈ M) of the Thompson series

Tg(q) =∞∑

n=−1

Tr(g|V \n)qn =

∞∑

n=−1

cg(n)qn.

It was pointed out in [JLW] that all the coefficients cg(n) are completely determinedif the values of ch(1), ch(2), ch(3), and ch(5) are known for all h ∈ M . In particular,the above relation is a complete recursive relation determining the coefficients c(n)of the elliptic modular function j(q)− 744.

FREE LIE ALGEBRAS 5

1. The Denominator Identity

Let Γ be a countable abelian semigroup such that every element α ∈ Γ canbe written as a sum of elements in Γ only in finitely many ways. For example,the semigroup Z>0 of positive integers satisfies the above condition, whereas themonoid Z≥0 of nonnegative integers doesn’t.

Let V =⊕

α∈Γ Vα be a Γ-graded vector space over C with dimVα < ∞ forall α ∈ Γ, and consider the free Lie algebra L = F(V ) generated by V . Foreach α ∈ Γ, let Lα be the subspace of L spanned by all the vectors of the form[[[· · · [v1, v2] · · · ]vk−1]vk] (k ≥ 1) with vi ∈ V such that degv1 + · · · + degvk = α.Then the free Lie algebra L has a Γ-gradation L =

⊕

α∈Γ Lα.

In this paper, we will use the denominator identity for the free Lie algebraL =

⊕

α∈Γ Lα to derive a closed form dimension formula for the homogeneoussubspaces Lα (α ∈ Γ) (cf. [Ka1], [Ju]), and discuss its applications to variousspecial cases. In particular, we will derive a number of interesting combinatorialidentities.

Let us consider first the general Γ-graded Lie algebras L =⊕

α∈Γ Lα such thatdimLα < ∞ for all α ∈ Γ. Let C be the trivial 1-dimensional L-module. Recallthat the homology modules Hk(L) = Hk(L,C) are determined from the followingcomplex

(1.1)· · · −→ Λk(L) dk−→ Λk−1(L) −→ · · ·

−→ Λ1(L) d1−→ Λ0(L) d0−→ C −→ 0,

where the differentials dk : Λk(L) → Λk−1(L) are defined by

(1.2) dk(x1 ∧ · · · ∧ xk) =∑

s<t

(−1)s+t[xs, xt] ∧ x1 ∧ · · · ∧ xs ∧ · · · ∧ xt ∧ · · · ∧ xk

for k ≥ 2, xi ∈ L, and d1 = d0 = 0.

For each k ≥ 0 and α ∈ Γ, we define Λk(L)α to be the subspace of Λk(L) spannedby the vectors of the form x1∧· · ·∧xk (xi ∈ L) such that deg(x1)+ · · ·+deg(xk) =α. Then the space Λk(L) has a Γ-gradation with finite dimensional homogeneoussubspaces, and hence the homology module Hk(L) inherits a Γ-gradation fromΛk(L). Therefore, by applying the Euler-Poincare principle to (1.1), we obtain

(1.3)∞∑

k=0

(−1)kchΛk(L) =∞∑

k=0

(−1)kchHk(L).

It is well-known that

(1.4)∞∑

k=0

(−1)kchΛk(L) =∏

α∈Γ

(1− eα)dimLα .


On the other hand, from the righthand side of (1.3), we define the homologyspace of the Γ-graded Lie algebra L to be an alternating direct sum of homologymodules:

(1.5) H =∞∑

k=1

(−1)k+1Hk(L) = H1(L)�H2(L)⊕H3(L)� · · · .

Then (1.3) can be written as

(1.6)∏

α∈Γ

(1− eα)dimLα = 1− chH.

We will call (1.6) the denominator identity for the Γ-graded Lie algebra L =⊕

α∈Γ Lα (cf. [KaKi]).

The identity (1.6) is a generalization of the usual denominator identity for sym-metrizable Kac-Moody algebras and generalized Kac-Moody algebras (cf. [K1],[GL], [B1]). For example, let g = g(A) be a symmetrizable Kac-Moody algebraassociated with a generalized Cartan matrix A = (aij)i,j∈I with the Weyl groupW , and let g− be the subalgebra of g generated by the negative root spaces. Weapply the identity (1.6) to the Lie algebra L = g−. By Kostant’s formula for thehomology modules over symmetrizable Kac-Moody algebras ([Ko], [GL]), we have

Hk(g−) =⊕

w∈Wl(w)=k

Cwρ−ρ

for k = 0, 1, 2, 3, · · · . Hence the righthand side of (1.6) is equal to

1− chH =∑

w∈W

(−1)l(w)ewρ−ρ,

which yields the denominator identity for symmetrizable Kac-Moody algebras

∏

α∈∆+

(1− e−α)dimgα =∑

w∈W

(−1)l(w)ewρ−ρ.

Now, let V =⊕

α∈Γ Vα be a Γ-graded vector space over C with dimVα = d(α) <∞ for all α ∈ Γ, and let L =

⊕

α∈Γ Lα be the free Lie algebra generated by V .Then, since H1(L) = V and Hk(L) = 0 for all k ≥ 2, the homology space H is equalto H = H1(L) = V . Therefore, the denominator identity for the free Lie algebra Lis the same as

(1.7)∏

α∈Γ

(1− eα)dimLα = 1− chV = 1−∑

α∈Γ

d(α)eα.

FREE LIE ALGEBRAS 7

We will derive a closed form dimension formula for the homogeneous subspaces Lα

of L from the denominator identity (1.7).

Let P (V ) = {α ∈ Γ| dimVα 6= 0}, and let P (V ) = {τ1, τ2, τ3, · · · } be an enumer-ation of P (V ). We denote by d(i) = dimVτi . For τ ∈ Γ, define

(1.8) T (τ) = {s = (si)i≥1| si ∈ Z≥0,∑

siτi = τ},

the set of all partitions of τ into a sum of τi’s. By our finiteness condition on thesemigroup Γ, the set T (τ) must be finite. For s ∈ T (τ), we will use the notation|s| =

∑

si and s! =∏

si! (cf. [Ju]). Now for τ ∈ Γ, we define a function

(1.9) W (τ) =∑

s∈T (τ)

(|s| − 1)!s!

∏

d(i)si .

We will call W (τ) the Witt partition function. Then, using the formal power se-ries log(1 − t) = −

∑∞k=1

tk

k and Mobius inversion, we obtain a nice closed formdimension formula for the homogeneous subspaces Lα (α ∈ Γ).

Theorem 1.1. (cf. [Ka1], [Ju]) Let V =⊕

α∈Γ Vα be a Γ-graded vector space overC, and let L =

⊕

α∈Γ Lα be the free Lie algebra generated by V . Then we have

(1.10) dimLα =∑

d|α

1dµ(d)W

(αd

)

=∑

d|α

1dµ(d)

∑

s∈T (α/d)

(|s| − 1)!s!

∏

d(i)si ,

where µ is the classical Mobius function, and, for a positive integer d, d|α denotesα = dτ for some τ ∈ Γ, in which case α/d = τ .

Proof. By the denominator identity (1.7), we have

∏

α∈Γ

(1− eα)−dimLα =1

1− chV=

11−

∑∞i=1 d(i)eτi

.

Using the formal power series log(1−t) = −∑∞

k=11k tk, we obtain from the lefthand

side

log

(

∏

α∈Γ

(1− eα)−dimLα

)

= −∑

α∈Γ

(dimLα)log(1− eα)

=∑

α∈Γ

(dimLα)∞∑

k=1

1k

ekα =∑

α∈Γ

∞∑

k=1

1k

(dimLα)ekα.


On the other hand, the righthand side yields

log(

11−

∑∞i=1 d(i)eτi

)

= −log

(

1−∞∑

i=1

d(i)eτi

)

=∞∑

k=1

1k

( ∞∑

i=1

d(i)eτi

)k

=∞∑

k=1

1k

∑

s=(si)si∈Z≥0P

si=k

(∑

si)!Π(si!)

(∏

d(i)si

)

eP

siτi

=∑

τ∈Γ

∑

s∈T (τ)

(|s| − 1)!s!

∏

d(i)si

eτ

=∑

τ∈Γ

W (τ)eτ .

Therefore, we have

W (τ) =∑

τ=kα

1k

(dimLα) =∑

k|τ

1k

dimL τk.

Hence, by Mobius inversion, we obtain

dimLα =∑

d|α

1dµ(d)W

(αd

)

. �

Remark. Even though the proof of the formula (1.10) can also be found in [Ka1]and [Ju], we include the proof here for the following reasons. First, the formula(1.10) can be extended to a dimension formula for any Γ-graded Lie algebras withalmost the same proof (cf. [KaKi]). Secondly, in the proof of Theorem 1.1, observethat the Witt partition function W (τ) is determined by the identity

(1.11)∑

τ∈Γ

W (τ)eτ = log(

11− chH

)

.

In many cases, we can use the identity (1.11) directly to obtain simple expressionsof the Witt partition function W (τ). Moreover, we will be able to derive a numberof combinatorial identities from various expressions of the Witt partition functionW (τ).

FREE LIE ALGEBRAS 9

2. The Witt Formula and its Generalization

Suppose V is an r-dimensional vector space over C, and let L = F(V ) be the freeLie algebra generated by V . For n ≥ 1, let Ln be the subspace of L spanned by allthe n-fold brackets [[· · · [v1, v2] · · · ]vn] with vi ∈ V . Then we have a Z>0-gradationon the free Lie algebra L = F(V ) =

⊕

n≥1 Ln. In particular, L1 = V . We willapply our dimension formula (1.10) to this setting. Since degv = 1 for all v ∈ V ,we have P (V ) = {1}, and dimV1 = dimV = r. Thus, for each n ≥ 1, since the onlypartition of n into a sum of 1’s is n = n(1), we have

(2.1) T (n) = {s = (n, 0, 0, · · · )},

and the Witt partition function W (n) simplifies to

(2.2) W (n) =(n− 1)!

n!rn =

1n

rn.

Indeed, using the identity (1.11), we have∞∑

n=1

W (n)qn = log(

11− chV

)

= log1

1− rq=

∞∑

n=1

1n

rnqn,

which yields W (n) =1n

rn.

Therefore, by our dimension formula (1.10), we obtain:

Proposition 2.1. (cf. [Bo], [J], [Se]) Suppose V is an r-dimensional vector spaceover C, and let L = F(V ) =

⊕∞n=1 Ln be the free Lie algebra generated by V . Then

we have

(2.3) dimLn =1n

∑

d|n

µ(d)rn/d.

Therefore, the denominator identity (1.7) yields a product formula

(2.4)∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)rn/d

= 1− rq.

Proof. By our dimension formula (1.10), we have

dimLn =∑

d|n

1dµ(d)W

(nd

)

=∑

d|n

1dµ(d)

dn

rn/d

=1n

∑

d|n

µ(d)rn/d. �


Remark. The formula (2.3) is called the Witt formula for the free Lie algebra onr generators ([Bo], [J], [Se]).

More generally, let V =⊕∞

i=1 Vi be a Z>0-graded vector space with dimVi =d(i) < ∞ for all i ≥ 1, and consider the free Lie algebra L = F(V ) generated byV . For n ≥ 1, let Ln be the subspace of L spanned by all the vector of the form[[[· · · [v1, v2] · · · ]vk−1]vk] (k ≥ 1) with vi ∈ V such that degv1 + · · · + degvk = n.Then the Lie algebra L has a Z>0-gradation L =

⊕∞n=1 Ln. We will apply our

dimension formula (1.10) to this setting. Note that P (V ) = Z>0 = {1, 2, 3, · · · },and for n ≥ 1, we have

(2.5) T (n) = {s = (si)i≥1| si ∈ Z≥0,∑

isi = n},

the set of all partitions of n into a sum of positive integers. Hence the Witt partitionfunction W (n) is equal to

(2.6) W (n) =∑

s∈T (n)

(|s| − 1)!s!

∏

d(i)si .

Therefore, by our dimension formula (1.10), we obtain the following “1-dimensionalgeneralization” of the Witt formula:

Theorem 2.2. Let V =⊕∞

i=1 Vi be a Z>0-graded vector space over C withdimVi = d(i) < ∞ for all i ≥ 1, and let L =

⊕∞n=1 Ln be the free Lie algebra

generated by V . Then we have

(2.7) dimLn =∑

d|n

1dµ(d)

∑

s∈T (n/d)

(|s| − 1)!s!

∏

d(i)si . �

Now, we will consider the applications of the dimension formula (2.7) to severalspecial cases. We first consider the case where d(i) = r for all i ≥ 1. In this case,the Witt partition function W (n) is equal to

(2.8) W (n) =∑

s∈T (n)

(|s| − 1)!s!

r|s|.

On the other hand, we can calculate W (n) directly using the identity (1.11). Notethat

chV = rq + rq2 + rq3 + · · · = rq1− q

.

FREE LIE ALGEBRAS 11

Thus we have

∞∑

n=1

W (n)qn = log(

11− chV

)

= log1

1− rq1− q

= log1− q

1− (r + 1)q= log(1− q)− log(1− (r + 1)q)

= −∞∑

n=1

1n

qn +∞∑

n=1

1n

(r + 1)nqn

=∞∑

n=1

1n

[(r + 1)n − 1]qn.

It follows that

(2.9) W (n) =1n

[(r + 1)n − 1].

Therefore, we obtain a combinatorial identity:

(2.10)∑

s∈T (n)

(|s| − 1)!s!

r|s| =1n

[(r + 1)n − 1].

In particular, if r = 1, we get

∑

s∈T (n)

(|s| − 1)!s!

=1n

(2n − 1).

Let k be a positive integer ≤ n. If we compare the coefficients of rk in both sidesof (2.10), we get

(2.11)∑

s∈T (n)|s|=k

(k − 1)!s!

=1n

(

nk

)

.

Note that in the lefthand side, the sum is taken over all the partitions of n into asum of k positive integers, i.e., over all the partitions of n into k parts.

Combining (2.9) with the dimension formula (2.7), we obtain:

Proposition 2.3. Let V =⊕∞

i=1 Vi be a Z>0-graded vector space over C withdimVi = r (i ≥ 1) for some r ∈ Z>0, and let L =

⊕∞n=1 Ln be the free Lie algebra


generated by V . Then we have

(2.12)

dimLn =1n

∑

d|n

µ(d)[(r + 1)n/d − 1]

=

1n

∑

d|n

µ(d)(r + 1)n/d if n 6= 1

r if n = 1.


(2.13)∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)[(r + 1)n/d − 1]

=1− (r + 1)q

1− q. �

Remark. The product formula (2.13) can also be deduced from (2.4) and (2.12).

Next, suppose d(i) = rai (i ≥ 1) for some r, a ∈ Z>0. Then, by (2.6) and (2.10),the Witt partition function W (n) simplifies to

(2.14)

W (n) =∑

s∈T (n)

(|s| − 1)!s!

∏

(rai)si

=∑

s∈T (n)

(|s| − 1)!s!

rP

siaP

isi = an∑

s∈T (n)

(|s| − 1)!s!

r|s|

=1n

an[(r + 1)n − 1].

Combining (2.14) with the dimension formula (2.7), we obtain:


i=1 Vi be a Z>0-graded vector space over C withdimVi = rai (i ≥ 1) for some r, a ∈ Z>0, and let L =

⊕∞n=1 Ln be the free Lie

algebra generated by V . Then we have

(2.15) dimLn =1n

∑

d|n

µ(d)an/d[(r + 1)n/d − 1].


(2.16)∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)an/d[(r + 1)n/d − 1]

=1− (r + 1)aq

1− aq. �

Remark. The product formula (2.16) can also be deduced from (2.4).

Combining (2.16) with (2.13), we obtain the following exchange principle:


Corollary 2.5.

(2.17)

∞∏

n=1

(1− anqn)

1n

∑

d|n

µ(d)[(r + 1)n/d − 1]

=∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)[(r + 1)n/d − 1]an/d

. �

Now, let us consider the case where d(1) = 0 and d(i) = a(i−1) (i ≥ 2) for somea ∈ Z>0. In this case, the Witt partition function W (n) is equal to

(2.18)

W (n) =∑

s∈T (n)

(|s| − 1)!s!

∏

[a(i− 1)]si

=∑

s∈T (n)

(|s| − 1)!s!

a|s|∏

(i− 1)si .

On the other hand, since

chV =∞∑

i=2

a(i− 1)qi = aq2∞∑

i=2

(i− 1)qi−2 = aq2∞∑

i=1

iqi−1

= aq2(

11− q

)′

=aq2

(1− q)2,

we have∞∑

n=1

W (n)qn = log(

11− chV

)

= log1

1− aq2

(1− q)2

= log(1− q)2

(1− q)2 − aq2 = log(1− q)2

[1− (1−√

a)q][1− (1 +√

a)q]= 2log(1− q)− log[1− (1−

√a)q]− log[1− (1 +

√a)q]

= −∞∑

n=1

2n

qn +∞∑

n=1

1n

(1−√

a)nqn +∞∑

n=1

1n

(1 +√

a)nqn

=∞∑

n=1

1n

(

(1 +√

a)n + (1−√

a)n − 2)

qn.

It follows that

(2.19) W (n) =1n

(

(1 +√

a)n + (1−√

a)n − 2)

.



(2.20)

∑

s∈T (n)

(|s| − 1)!s!

a|s|∏

(i− 1)si =1n

(

(1 +√

a)n + (1−√

a)n − 2)

=2n

[ n2 ]

∑

k=1

(

n2k

)

ak.

If a = 1, then we get∑

s∈T (n)

(|s| − 1)!s!

∏

(i− 1)si =2n

(2n−1 − 1).

By (2.10), we have

2n−1 − 1 = (n− 1)∑

t∈T (n−1)

(|t| − 1)!t!

.

It follows that

(2.21)

∑

s∈T (n)

(|s| − 1)!s!

∏

(i− 1)si =2(n− 1)

n

∑

t∈T (n−1)

(|t| − 1)!t!

=2n

(2n−1 − 1),

which relates a Witt partition function associated with the partitions of n and aWitt partition function associated with the partitions of n− 1.

Moreover, let k be a positive integer ≤ n. Then, by comparing the coefficientsof ak of both sides of (2.20), we get

(2.22)∑

s∈T (n)|s|=k

(k − 1)!s!

∏

(i− 1)si =2n

(

n2k

)

.

For k ≤ n2 , combining (2.22) with (2.11), we obtain

(2.23)∑

s∈T (n)|s|=k

(k − 1)!s!

∏

(i− 1)si = 2∑

s∈T (n)|s|=2k

(2k − 1)!s!

.

In particular, if k > [n2 ], then

∑

s∈T (n)|s|=k

(k − 1)!s!

∏

(i− 1)si = 0.

Indeed, if s ∈ T (n) and |s| = k > [n2 ], then we must have s1 > 0, and hence

∏

(i− 1)si = 0, which implies the above identity.

Now, combining (2.19) with the dimension formula (2.7), we obtain:



i=1 Vi be a Z>0-graded vector space over C withdimV1 = 0 and dimVi = a(i − 1) (i ≥ 2) for some a ∈ Z>0, and let L =

⊕∞n=1 Ln

be the free Lie algebra generated by V . Then we have

(2.24)

dimLn =1n

∑

d|n

µ(d)(

(1 +√

a)n/d + (1−√

a)n/d − 2)

=

1n

∑

d|n

µ(d)(

(1 +√

a)n/d + (1−√

a)n/d)

if n 6= 1

0 if n = 1.


(2.25)

∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)(

(1 +√

a)n/d + (1−√

a)n/d − 2)

=(1− q)2 − aq2

(1− q)2. �

Remark. The product formula (2.25) can also be deduced from (2.4) and (2.24).When a = 2, ε = 1 +

√2 is the fundamental unit of Q(

√2) and (2.25) can be

rewritten as

(2.26)∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)Tr(εn/d)

= 1− 2q − q2,

where Tr is the trace map from Q(√

2) to Q.

Finally, suppose d(1) = 0 and d(i) = aα2(i − 1)βi−2 (i ≥ 2) for some positiveintegers a, α, and β. In this case, the Witt partition function W (n) is equal to

(2.27)

W (n) =∑

s∈T (n)

(|s| − 1)!s!

∏[

aα2(i− 1)βi−2]si

=∑

s∈T (n)

(|s| − 1)!s!

a|s|α2|s|βn−2|s|∏

(i− 1)si

= βn∑

s∈T (n)

(|s| − 1)!s!

(

aα2

β2

)|s|∏

(i− 1)si .


On the other hand, note that

chV =∞∑

i=2

aα2(i− 1)βi−2qi = aα2q2∞∑

i=2

(i− 1)(βq)i−2

= aα2q2∞∑

i=1

i(βq)i−1 = aα2q2 1β

(

11− βq

)′

=aα2q2

(1− βq)2.

It follows that∞∑

n=1

W (n)qn = log(

11− chV

)

= log1

1− aα2q2

(1− βq)2

= log(1− βq)2

(1− βq)2 − aα2q2 = log(1− βq)2

(1− (β + α√

a)q)(1− (β − α√

a)q)

= 2log(1− βq)− log(1− (β + α√

a)q)− log(1− (β − α√

a)q)

= −∞∑

n=1

2n

βnqn +∞∑

n=1

1n

(β + α√

a)nqn +∞∑

n=1

1n

(β − α√

a)nqn

=∞∑

n=1

1n

[

(β + α√

a)n + (β − α√

a)n − 2βn]

qn,

which yields

(2.28) W (n) =1n

[

(β + α√

a)n + (β − α√

a)n − 2βn]

.


(2.29)βn

∑

s∈T (n)

(|s| − 1)!s!

(

aα2

β2

)|s|∏

(i− 1)si

=1n

[

(β + α√

a)n + (β − α√

a)n − 2βn]

.

Now, combining (2.28) with the dimension formula (2.7), we obtain:


i=1 Vi be a Z>0-graded vector space over C withdimV1 = 0 and dimVi = aα2(i − 1)βi−2 (i ≥ 2) for some a, α, β ∈ Z>0, and letL =

⊕∞n=1 Ln be the free Lie algebra generated by V . Then we have

(2.30) dimLn =1n

∑

d|n

µ(d)[

(β + α√

a)n/d + (β − α√

a)n/d − 2βn/d]

.



(2.31)

∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)[

(β + α√

a)n/d + (β − α√

a)n/d − 2βn/d]

=(1− βq)2 − aα2q2

(1− βq)2. �

Remark. (a) The product formula (2.31) can also be deduced from (2.4).

(b) Let a be a square free positive integer and let γ = β + α√

a ∈ Z[√

a]. Then(2.31) can be rewritten as

(2.32)∞∏

n=1

(1− qn)

1n

∑

d|n

µ(d)Tr(γn/d)

= 1− Tr(γ)q + N(γ)q2,

which generalizes (2.26). Here, Tr and N are the trace and the norm maps fromQ(√

a) to Q.

We collect the combinatorial identities we obtained in this section in the followingproposition.

Proposition 2.8.

(a)∑

s∈T (n)

(|s| − 1)!s!

r|s| =1n

[(r + 1)n − 1],

∑

s∈T (n)

(|s| − 1)!s!

=1n

(2n − 1).

(b)∑

s∈T (n)|s|=k

(k − 1)!s!

=1n

(

nk

)

.

∑

s∈T (n)

(|s| − 1)!s!

a|s|∏

(i− 1)si =1n

(

(1 +√

a)n + (1−√

a)n − 2)

(c)

=2n

[ n2 ]

∑

k=1

(

n2k

)

ak.


∑

s∈T (n)

(|s| − 1)!s!

∏

(i− 1)si =2(n− 1)

n

∑

t∈T (n−1)

(|t| − 1)!t!

(d)

=2n

(2n−1 − 1).

∑

s∈T (n)|s|=k

(k − 1)!s!

∏

(i− 1)si = 2∑

s∈T (n)|s|=2k

(2k − 1)!s!

(e)

=2n

(

n2k

)

for 1 ≤ k ≤ n2

.

βn∑

s∈T (n)

(|s| − 1)!s!

(

aα2

β2

)|s|∏

(i− 1)si(f)

=1n

[

(β + α√

a)n + (β − α√

a)n − 2βn]

. �


3. Further Generalization

In this section, we consider Z>0 × Z>0-graded vector spaces V =⊕∞

i,j=1 V(i,j)

with dimV(i,j) = d(i, j) < ∞ for all i, j ≥ 1. Let L = F(V ) be the free Lie algebragenerated by V . For m,n ≥ 1, let L(m,n) be the subspace of L spanned by allthe vectors of the form [[[· · · [v1, v2] · · · ]vk−1]vk] (k ≥ 1) with vi ∈ V such thatdegv1 + · · · + degvk = (m,n). Then the Lie algebra L has a Z>0 × Z>0-gradationL =

⊕∞m,n=1 L(m,n). We will apply our dimension formula (1.10) to this setting.

Note thatP (V ) = Z>0 × Z>0 = {(i, j)| i, j = 1, 2, 3, · · · },

and for m,n ≥ 1, we have

(3.1) T (m,n) = {s = (sij)i,j≥1| si,j ∈ Z≥0,∑

sij(i, j) = (m,n)},

which is the set of all partitions of (m,n) into a sum of ordered pairs of positiveintegers. Hence the Witt partition function W (m,n) is equal to

(3.2) W (m,n) =∑

s∈T (m,n)

(|s| − 1)!s!

∏

d(i, j)sij ,

where |s| =∑

sij and s! =∏

sij !. Therefore, our dimension formula (1.10) yieldsthe following “2-dimensional generalization” of the Witt formula:

Theorem 3.1. Let V =⊕∞

i,j=1 V(i,j) be a Z>0 ×Z>0-graded vector space over Cwith dimV(i,j) = d(i, j) < ∞ for all i, j ≥ 1, and let L =

⊕∞m,n=1 L(m,n) be the free

Lie algebra generated by V . Then we have

(3.3) dimL(m,n) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

∏

d(i, j)sij . �

Now, we will consider the applications of the dimension formula (3.3) to variousspecial cases. We first consider the case where d(i, j) = r (i, j ≥ 1) for some positiveinteger r. In this case, the Witt partition function W (m,n) is equal to

(3.4) W (m,n) =∑

s∈T (m,n)

(|s| − 1)!s!

r|s|.

On the other hand, since

chV =∞∑

i,j=1

rpiqj = rpq(∞∑

i=0

pi)(∞∑

j=0

qj) =rpq

(1− p)(1− q),


we have

∞∑

m,n=1

W (m,n)pmqn = log(

11− chV

)

= log1

1− rpq(1− p)(1− q)

= log(1− p)(1− q)

(1− p)(1− q)− rpq= log

(1− p)(1− q)1− [p + q + (r − 1)pq]

= log(1− p) + log(1− q)− log (1− [p + q + (r − 1)pq])

= −∞∑

k=1

1k

pk −∞∑

k=1

1k

qk +∞∑

k=1

1k

[p + q + (r − 1)pq]k

= −∞∑

k=1

1k

pk −∞∑

k=1

1k

qk +∞∑

k=1

1k

∑

a+b+c=ka,b,c≥0

k!a!b!c!

paqb(r − 1)cpcqc

=∞∑

k=1

1k

∑

a+b+c=ka,b,c≥0

a6=k,b 6=k

k!a!b!c!

(r − 1)cpa+cqb+c

=∞∑

m,n=1

min(m,n)∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(r − 1)c

pmqn,

which yields

(3.5) W (m,n) =min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(r − 1)c.

Therefore, we obtain a combinatorial identity

(3.6)∑

s∈T (m,n)

(|s| − 1)!s!

r|s| =min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(r − 1)c.

If r = 1, we get

∑

s∈T (m,n)

(|s| − 1)!s!

=(m + n− 1)!

m! n!=

1m + n

(

m + nn

)

.

It follows from (2.11) that

1m + n

(

m + nn

)

=∑

t∈T (m+n)|t|=n

(n− 1)!t!

=∑

t∈T (m+n)|t|=m

(m− 1)!t!

.


Therefore, we obtain a combinatorial identity, which relates a Witt partition func-tion associated with the partitions of an ordered pair of positive integers into a sumof ordered pairs of positive integers and a Witt partition function associated withthe partitions of a positive integer into a sum of positive integers:

(3.7)

∑

s∈T (m,n)

(|s| − 1)!s!

=∑

t∈T (m+n)|t|=n

(n− 1)!t!

=∑

t∈T (m+n)|t|=m

(m− 1)!t!

=1

m + n

(

m + nn

)

.

If we compare the coefficients of rk (k ≤ m,n) of both sides of (3.6), we getanother combinatorial identity:

(3.8)

∑

s∈T (m,n)|s|=k

(k − 1)!s!

=min(m,n)

∑

c=k

(m + n− c− 1)!(m− c)!(n− c)!c!

(

ck

)

(−1)c−k

=min(m,n)

∑

c=k

1m + n− c

(

m + n− cn

)(

nc

)(

ck

)

(−1)c−k.

Suppose m ≥ n, and let k = n. Then (3.8) yields

∑

s∈T (m,n)|s|=n

(n− 1)!s!

=(m− 1)!

(m− n)!n!=

1m

(

mn

)

.

Moreover, if m > n, by (2.11) and (3.7), we have

1m

(

mn

)

=∑

t∈T (m)|t|=n

(n− 1)!t!

=∑

s∈T (m−n,n)

(|s| − 1)!s!

.

Therefore, if m > n, we obtain the following combinatorial identity:

(3.9)

∑

s∈T (m,n)|s|=n

(n− 1)!s!

=∑

t∈T (m)|t|=n

(n− 1)!t!

=∑

s∈T (m−n,n)

(|s| − 1)!s!

=1m

(

mn

)

.


Example 3.2. Let m = 5, n = 3. Then the partitions s ∈ T (5, 3) with |s| = 3 are

(5, 3) = (1, 1) + (1, 1) + (3, 1)

= (1, 1) + (2, 1) + (2, 1),

and they are in 1-1 correspondence with

5 = 1 + 1 + 3

= 1 + 2 + 2,

the partitions of 5 into 3 parts. On the other hand, the partitions s of (2, 3) are

(2, 3) = (2, 3)

= (1, 1) + (1, 2).

It is straightforward to check that

∑

s∈T (5,3)|s|=3

2!s!

=∑

t∈T (5)|t|=3

2!t!

=∑

s∈T (2,3)

(|s| − 1)!s!

=15

(

53

)

= 2. �

Now combining (3.4) and (3.5) with the dimension formula (3.3), we obtain:


i,j=1 V(i,j) be a Z>0 × Z>0-graded vector spaceover C with dimV(i,j) = r (i, j ≥ 1) for some positive integer r, and let L =⊕∞

m,n=1 L(m,n) be the free Lie algebra generated by V . Then we have

(3.10)

dimL(m,n) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

r|s|

=∑

d|(m,n)

1dµ(d)

min(m/d,n/d)∑

c=0

(md + n

d − c− 1)!(m

d − c)!(nd − c)!c!

(r − 1)c.

Therefore, by letting l(m, n) = dimL(m,n) for all m, n ≥ 1, the denominator identity(1.7) yields a product formula

(3.11)∞∏

m,n=1

(1− pmqn)l(m,n) =1− [p + q + (r − 1)pq]

(1− p)(1− q). �


Next, we consider the case where d(i, j) = raibj (i, j ≥ 1) for some positiveintegers r, a, and b. In this case, the Witt partition function W (m,n) is equal to

(3.12)

W (m,n) =∑

s∈T (m,n)

(|s| − 1)!s!

∏

(raibj)sij

= ambn

∑

s∈T (m,n)

(|s| − 1)!s!

r|s|

.

By (3.6), we get

(3.13) W (m,n) = ambnmin(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(r − 1)c.

Combining (3.12) and (3.13) with the dimension formula (3.3), we obtain:


i,j=1 V(i,j) be a Z>0×Z>0-graded vector space overC with dimV(i,j) = raibj (i, j ≥ 1) for some positive integers r, a, and b, and letL =

⊕∞m,n=1 L(m,n) be the free Lie algebra generated by V . Then we have

(3.14)

dimL(m,n) =∑

d|(m,n)

1dµ(d)am/dbn/d

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

r|s|

=∑

d|(m,n)

1dµ(d)am/dbn/d

min(m/d,n/d)∑

c=0

(md + n

d − c− 1)!(m

d − c)!(nd − c)!c!

(r − 1)c.


(3.15)∞∏

m,n=1

(1− pmqn)l(m,n) =1− [ap + bq + (r − 1)abpq]

(1− ap)(1− bq). �

Comparing (3.11) and (3.15), we obtain the following double exchange principlewhich is an analogue of (2.17):

Corollary 3.5.

(3.16)

∞∏

m,n=1

(1− ampmbnqn)

∑

d|(m,n)

1dµ(d)Sr(m/d, n/d)

=∞∏

m,n=1

(1− pmqn)

∑

d|(m,n)

1dµ(d)Sr(m/d, n/d)am/dbn/d

,


where

Sr(m, n) =∑

s∈T (m,n)

(|s| − 1)!s!

r|s|

=min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(r − 1)c. �

Finally, suppose d(i, j) = a(i− 1)(j− 1)αiβj (i, j ≥ 2) for some positive integersa, α, and β. Then the Witt partition function W (m,n) is equal to

(3.17)

W (m,n) =∑

s∈T (m,n)

(|s| − 1)!s!

∏

[a(i− 1)(j − 1)αiβj ]sij

=∑

s∈T (m,n)

(|s| − 1)!s!

a|s|αmβn∏

(i− 1)sij (j − 1)sij

= αmβn

∑

s∈T (m,n)

(|s| − 1)!s!

a|s|∏


.

On the other hand, note that

chV =∞∑

i,j=2

a(i− 1)(j − 1)(αp)i(βq)j

= a(αp)2(βq)2∞∑

i,j=2

(i− 1)(j − 1)(αp)i−2(βq)j−2

= a(αp)2(βq)2[ ∞∑

i=1

i(αp)i−1

]

∞∑

j=1

j(βq)j−1

= a(αp)2(βq)21α

(

11− αp

)′ 1β

(

11− βq

)′

=a(αp)2(βq)2

(1− αp)2(1− βq)2.


Therefore, we have

∞∑

m,n=1

W (m,n)pmqn = log(

11− chV

)

= log1

1− a(αp)2(βq)2

(1− αp)2(1− βq)2

= log(1− αp)2(1− βq)2

(1− αp)2(1− βq)2 − a(αp)2(βq)2

= log(1− αp)2(1− βq)2

[(1− αp)(1− βq) +√

aαβpq][(1− αp)(1− βq)−√

aαβpq]

= log(1− αp)2(1− βq)2

[1− (αp + βq − (√

a + 1)αβpq)][1− (αp + βq + (√

a− 1)αβpq)]

= 2log(1− αp) + 2log(1− βq)

− log[1− (αp + βq − (√

a + 1)αβpq)]

− log[1− (αp + βq + (√

a− 1)αβpq)]

= −2∞∑

k=1

1k

(αp)k − 2∞∑

k=1

1k

(βq)k

+∞∑

k=1

1k

∑

r+s+t=kr,s,t≥0

k!r!s!t!

(αp)r(βq)s(−1)t(√

a + 1)t(αp)t(βq)t

+∞∑

k=1

1k

∑

r+s+t=kr,s,t≥0

k!r!s!t!

(αp)r(βq)s(√

a− 1)t(αp)t(βq)t

=∞∑

k=1

1k

∑

r+s+t=kr,s,t≥0

r 6=k,s 6=k

k!r!s!t!

(−1)t(√

a + 1)t(αp)r+t(βq)s+t

+∞∑

k=1

1k

∑

r+s+t=kr,s,t≥0

r 6=k,s 6=k

k!r!s!t!

(√

a− 1)t(αp)r+t(βq)s+t

=∞∑

m,n=1

min(m,n)∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(−1)c(√

a + 1)c

αmβnpmqn

+∞∑

m,n=1

min(m,n)∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(√

a− 1)c

αmβnpmqn

=∞∑

m,n=1

αmβn

min(m,n)∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

((√

a− 1)c + (−1)c(√a + 1)c)

pmqn.


It follows that

(3.18) W (m,n) = αmβnmin(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(

(√

a− 1)c + (−1)c(√

a + 1)c),

which yields a combinatorial identity:

(3.19)

∑

s∈T (m,n)

(|s| − 1)!s!

a|s|∏


=min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(

(√

a− 1)c + (−1)c(√

a + 1)c).

If a = 1, we get

(3.20)

∑

s∈T (m,n)

(|s| − 1)!s!

∏


=min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(−2)c.

Moreover, if we compare the coefficients of ak (k ≤ m,n) of both sides of (3.19),we get

(3.21)

∑

s∈T (m,n)|s|=k

(k − 1)!s!

∏


= 2min(m,n)

∑

c=2k

(m + n− c− 1)!(m− c)!(n− c)!c!

(

c2k

)

(−1)c

= 2min(m,n)

∑

c=2k

1m + n− c

(

m + n− cn

)(

nc

)(

c2k

)

(−1)c.

For k ≤ n2 , combining (3.21) with (3.8), we obtain

(3.22)∑

s∈T (m,n)|s|=k

(k − 1)!s!

∏

(i− 1)sij (j − 1)sij = 2∑

s∈T (m,n)|s|=2k

(2k − 1)!s!

.

Observe that∑

s∈T (m,n)|s|>[ n

2 ]

(|s| − 1)!s!

∏

(i− 1)sij (j − 1)sij = 0.


In particular, if k = 0, we get

(3.23)

min(m,n)∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(−1)c

=min(m,n)

∑

c=0

1m + n− c

(

m + n− cn

)(

nc

)

(−1)c = 0,

which can also be deduced from (3.8). Moreover, if m ≥ n, then by comparing thecoefficients of ak with k = [n

2 ] in (3.21), we obtain

(3.24)∑

s∈T (m,n)|s|=[ n

2 ]

([n2 ]− 1)!

s!

∏

(i−1)sij (j−1)sij =

2n

(

m− 1n− 1

)

if n = 2k

2(

m− 1n− 2

)

if n = 2k + 1.

Note that (3.24) can also be deduced from (3.8) and (3.22).

Now, combining (3.17) and (3.18) with the dimension formula (3.3), we obtain:


i,j=1 V(i,j) be a Z>0×Z>0-graded vector space overC with dimV(i,j) = a(i − 1)(j − 1)αiβj (i, j ≥ 1) for some positive integers a, α,and β, and let L =

⊕∞m,n=1 L(m,n) be the free Lie algebra generated by V . Then

we have

dimL(m,n) =∑

d|(m,n)

1dµ(d)αm/dβn/d

×

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

a|s|∏


(3.25)

=∑

d|(m,n)

1dµ(d)α

md β

nd

min( md , n

d )∑

c=0

(md + n

d − c− 1)!(m

d − c)!(nd − c)!c!

(

(√

a− 1)c + (−1)c(√

a + 1)c).


(3.26)∞∏

m,n=1

(1− pmqn)l(m,n) =(1− αp)2(1− βq)2 − a(αp)2(βq)2

(1− αp)2(1− βq)2. �

Remark. The product formula (3.26) can also be deduced from (3.15).

We collect the combinatorial identities we obtained in this section in the followingproposition.


Proposition 3.7.

(a)∑

s∈T (m,n)

(|s| − 1)!s!

r|s| =min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(r − 1)c.

∑

s∈T (m,n)

(|s| − 1)!s!

=∑

t∈T (m+n)|t|=n

(n− 1)!t!

(b)

=∑

t∈T (m+n)|t|=m

(m− 1)!t!

=1

m + n

(

m + nn

)

.

∑

s∈T (m,n)|s|=k

(k − 1)!s!

=min(m,n)

∑

c=k

(m + n− c− 1)!(m− c)!(n− c)!c!

(

ck

)

(−1)c−k(c)

=min(m,n)

∑

c=k

1m + n− c

(

m + n− cn

)(

nc

)(

ck

)

(−1)c−k.

∑

s∈T (m,n)|s|=n

(n− 1)!s!

=∑

t∈T (m)|t|=n

(n− 1)!t!

=1m

(

mn

)

(m ≥ n)(d)

=∑

s∈T (m−n,n)

(|s| − 1)!s!

(m > n).

∑

s∈T (m,n)

(|s| − 1)!s!

a|s|∏

(i− 1)sij (j − 1)sij(e)

=min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(

(√

a− 1)c + (−1)c(√

a + 1)c).

∑

s∈T (m,n)

(|s| − 1)!s!

∏

(i− 1)sij (j − 1)sij(f)

=min(m,n)

∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(−2)c.


∑

s∈T (m,n)|s|=k

(k − 1)!s!

∏

(i− 1)sij (j − 1)sij(g)

= 2min(m,n)

∑

c=2k

(m + n− c− 1)!(m− c)!(n− c)!c!

(

c2k

)

(−1)c

= 2min(m,n)

∑

c=2k

1m + n− c

(

m + n− cn

)(

nc

)(

c2k

)

(−1)c.

= 2∑

s∈T (m,n)|s|=2k

(2k − 1)!s!

for 1 ≤ k ≤ n2

.

min(m,n)∑

c=0

(m + n− c− 1)!(m− c)!(n− c)!c!

(−1)c(h)

=min(m,n)

∑

c=0

1m + n− c

(

m + n− cn

)(

nc

)

(−1)c = 0.

(i)

∑

s∈T (m,n)|s|=[ n

2 ]

([n2 ]− 1)!

s!

∏

(i− 1)sij (j − 1)sij =

2n

(

m− 1n− 1

)

if n is even ≤ m

2(

m− 1n− 2

)

if n is odd ≤ m. �


4. Product Formula for Formal Power Series

In this section, we will discuss the relation of the free Lie algebras and theproduct formulas for formal power series. Let Γ be a countable abelian semigroupsatisfying the finiteness condition given in Section 1. Consider a formal power series

(4.1) 1−∑

α∈Γ

d(α)eα with d(α) ∈ Z>0 for all α ∈ Γ.

Suppose we have a product formula for the above formal power series

(4.2)∏

α∈Γ

(1− eα)c(α) = 1−∑

α∈Γ

d(α)eα

for some positive integers c(α). Then the identity (4.2) can be interpreted as thedenominator identity for a suitably defined free Lie algebra.

More precisely, let V =⊕

α∈Γ Vα be a Γ-graded vector space over C withdimVα = d(α) for all α ∈ Γ. Then the righthand side of (4.2) can be interpreted as1− chV . Let L be the free Lie algebra generated by V . As we have seen in Section1, the Lie algebra L has a Γ-gradation L =

⊕

α∈Γ Lα induced by that of V , andthe denominator identity (1.7) for the free Lie algebra L can be written as

(4.3)∏

α∈Γ

(1− eα)dimLα = 1− chV = 1−∑

α∈Γ

d(α)eα.

Therefore, the identity (4.2) is equal to the denominator identity for the free Liealgebra L =

⊕

α∈Γ Lα. In particular, we have

(4.4) dimLα = c(α) for all α ∈ Γ.

On the other hand, let {τ1, τ2, τ3, · · · } be an enumeration of Γ, and we denoteby d(i) = d(τi) = dimVτi for all i = 1, 2, 3, · · · . Then we can apply our dimen-sion formula (1.10) to the free Lie algebra L =

⊕

α∈Γ Lα, which would yield acombinatorial identity:

(4.5) c(α) =∑

d|α

1dµ(d)

∑

s∈T (α/d)

(|s| − 1)!s!

∏

d(i)si .

As an application, let us consider the elliptic modular function j. We write

(4.6) j(q)− 744 =∞∑

n=−1

c(n)qn = q−1 + 196884q + 21493760q2 + · · · .


Hence c(−1) = 1, c(0) = 0, and c(1) = 196884, etc.

It is shown in [B2] that the modular function j satisfies a product formula

(4.7) p−1∏

m>0n∈Z

(1− pmqn)c(mn) = j(p)− j(q).

Observing that c(0) = 0 and c(−k) = 0 for k > 1, (4.7) can be rewritten as

(4.8)∞∏

m,n=1

(1− pmqn)c(mn) =j(p)− j(q)p−1 − q−1 .

We expand the righthand side of (4.8) as follows:

j(p)− j(q)p−1 − q−1 =

∑∞n=−1 c(n)(pn − qn)

p−1 − q−1

=p−1 − q−1 +

∑∞n=1 c(n)(pn − qn)

p−1 − q−1

= 1− pq∞∑

n=1

c(n)pn − qn

p− q

= 1− pq∞∑

n=1

c(n)(pn−1 + pn−2q + · · ·+ qn−1)

= 1−∞∑

n=1

n∑

k=1

c(n)pn−k+1qk

= 1−∞∑

i,j=1

c(i + j − 1)piqj .

Therefore we obtain a product formula

(4.9)∞∏

m,n=1

(1− pmqn)c(mn) = 1−∞∑

i,j=1

c(i + j − 1)piqj .

Now, let V =⊕∞

i,j=1 V(i,j) be a Z>0 × Z>0-graded vector space over C withdimV(i,j) = c(i + j − 1) for all i, j ≥ 1, and let L be the free Lie algebra generatedby V . Then the Lie algebra L has a Z>0 × Z>0-gradation L =

⊕∞m,n=1 L(m,n)

induced by that of V , and the identity (4.9) is the denominator identity for the freeLie algebra L. Therefore, we have

(4.10) dimL(m,n) = c(mn) for all m,n ≥ 1.


On the other hand, we can apply our dimension formula (1.10) to the Z>0×Z>0-graded free Lie algebra L =

⊕∞m,n=1 L(m,n). In this case, recall that

P (V ) = Z>0 × Z>0 = {(i, j)| i, j = 1, 2, 3, · · · },

and for m,n ≥ 1, we have

T (m,n) = {s = (sij)i,j≥1| si,j ∈ Z≥0,∑

sij(i, j) = (m,n)},

which is the set of all partitions of (m,n) into a sum of ordered pairs of positiveintegers (cf. Section 3). Hence the Witt partition function W (m,n) is equal to

(4.11) W (m,n) =∑

s∈T (m,n)

(|s| − 1)!s!

∏

c(i + j − 1)sij .

Therefore, our dimension formula (1.10) yields:


i,j=1 V(i,j) be a Z>0×Z>0-graded vector space overC with dimV(i,j) = c(i + j− 1) for all i, j ≥ 1, where the c(n) are the coefficients ofthe elliptic modular function

j(q)− 744 =∞∑

n=−1

c(n)qn = q−1 + 196884q + 21493760q2 + · · · .

Let L =⊕∞

m,n=1 L(m,n) be the free Lie algebra generated by V . Then we have

(4.12) dimL(m,n) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

∏

c(i + j − 1)sij . �

Combining (4.12) with (4.10), we obtain the following interesting relations forthe coefficients c(n) of the elliptic modular function j(q)− 744.

Corollary 4.2.

(4.13) c(mn) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

∏

c(i + j − 1)sij . �

Remark. The relation (4.13) was obtained independently by Kang [Ka2] and Ju-risich [Ju] using the theory of generalized Kac-Moody algebras. Let M be theMonster simple group and let V \ =

⊕∞n=−1 V \

n be the Moonshine Module for theMonster simple group M ([FLM]). Then, in [JLW] and [KK], the relation (4.13)


was generalized to the relations of the coefficients cg(n) (g ∈ M) of the Thompsonseries

Tg(q) =∞∑

n=−1

Tr(g|V \n)qn =

∞∑

n=−1

cg(n)qn.

More precisely, for m, n > 0, we have

(4.14) cg(mn) =∑

d|(m,n)

1dµ(d)

∑

s∈T (m/d,n/d)

(|s| − 1)!s!

∏

cgd(i + j − 1)sij .

It was pointed out in [JLW] that these relations completely determine all the coef-ficients cg(n) if the values of ch(1), ch(2), ch(3), and ch(5) are known for all h ∈ M .In particular, the relation (4.13) is a complete recursive relation determining thecoefficients c(n) of the elliptic modular function j(q)− 744.


References

[B1] Borcherds, R.E., Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501–512.

[B2] Borcherds, R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math.109 (1992), 405–444.

[Bo] Bourbaki, N., Lie Groups and Lie Algebras, Part 1, Herman, Paris, 1975.

[CE] Cartan, H., Eilenberg, S., Homological Algebra, Princeton University Press, 1956.

[FLM] Frenkel, I.B., Lepowsky, J., Meurman, A., Vertex Operator Algebras and the Monster,Academic Press, 1988.

[GL] Garland, H., Lepowsky, J., Lie algebra homology and the Macdonald-Kac formulas,Invent. Math. 34 (1976), 37–76.

[J] Jacobson, N., Lie Algebras, 2nd ed., Dover, New York, 1979.

[Ju] Jurisich, E., Generalized Kac-Moody Lie algebras, free Lie algebras, and the structureof the monster Lie algebra (to appear in J. Pure and Applied Algebra).

[JLW] Jurisich, E., Lepowsky, J., Wilson, R.L., Realizations of the Monster Lie algebra,preprint (1994).

[K1] Kac, V.G., Infinite-dimensional Lie algebras and Dedekind’s η-function, Funct. Anal.Appl. 8 (1974), 68–70.

[K2] Kac, V.G., Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press,1990.

[KK] Kac, V.G., Kang, S.-J., Trace formula for graded Lie algebras and Monstrous Moon-shine, RIM-GARC preprint 94-30 (1994).

[Ka1] Kang, S.-J., Kac-Moody Lie algebras, Spectral sequences, and the Witt formula, Trans.Amer. Math. Soc. 339 (1993), 463–495.

[Ka2] Kang, S.-J., Generalized Kac-Moody algebras and the modular function j, Math. Ann.298 (1994), 373–384.

[KaKi] Kang, S.-J., Kim, M.-H., Dimension formula for graded Lie algebras and its applica-tions, to appear.

[Ko] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann.Math. 74 (1961), 329–387.

[Se] Serre, J.P., Lie Algebras and Lie Groups, Benjamin, New York, 1965.

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