A Recent Look at the Quantum Berezinian in the Yangian Y (𝔤𝔩_{m|n})

Lucy Gow^{∗}

December 28, 2004

Brundan and Kleshchev recently introduced a new family of presentations of the Yangian Y (𝔤𝔩_{n}) associated to the general linear Lie algebra 𝔤𝔩_{n}, and thus provided a fresh approach to its study. In this article, we would like to show how some of their ideas can be fruitfully extended to consider the Yangian Y (𝔤𝔩_{m|n}) associated to the Lie superalgebra 𝔤𝔩_{m|n}. In particular, we give a new proof of the result by Nazarov that the quantum Berezinian is central.^{1}

1 Definition of Y (𝔤𝔩_{m|n})

The Yangian Y (𝔤𝔩_{m|n}) is defined in [7] to be the ℤ_{2}-graded associative algebra over ℂ with generators t_{ij}^{(r)} and certain relations described below. We define the formal power series

t_{ij}(u) = δ_{ij} + t_{ij}^{(1)}u^{−1} + t_{ij}^{(2)}u^{−2} + …,

and a matrix

T (u) = ∑_{i,j=1}^{m+n} t_{ij}(u) ⊗ E_{ij} (−1)^{\overline{j}(\overline{i}+1)},  (1)

where E_{ij} is the standard elementary matrix and \overline{i} is the parity of the index i. In analogy with the usual Yangian Y (𝔤𝔩_{n}) (see for example [2], [5], [6]), the defining relations are then expressed by the matrix product

R(u − v)T_{1}(u)T_{2}(v) = T_{2}(v)T_{1}(u)R(u − v)

where

R(u − v) = 1 − \frac{1}{(u − v)}P_{12}

∗lucyg@maths.usyd.edu.au

1The author would like to acknowledge the guidance and support of her PhD supervisor Alexander Molev.

and P_{12} is the permutation matrix:

P_{12} = ∑_{i,j=1}^{m+n} E_{ij} ⊗ E_{ji}(−1)\overline{j}.

Then we have the following equivalent form of the defining relations:

[t_{ij}(u), t_{kl}(v)] = \frac{(−1)^{\overline{i}\overline{j}+\overline{i}\overline{k}+\overline{j}\overline{k}}}{(u − v)} (t_{kj}(u)t_{il}(v) − t_{kj}(v)t_{il}(u)).

Throughout this article we will observe the following notation for entries of the inverse matrix of T (u):

T (u)^{−1} =: (t′_{ij}(u))_{i,j=1}^{n} .

A straightforward calculation yields the following relation in Y (𝔤𝔩_{m|n}):

[t_{ij}(u), t′_{kl}(v)] = \frac{(−1)^{\overline{i}\overline{j}+\overline{i}\overline{k}+\overline{j}\overline{k}}}{(u − v)} · ( δ_{kj} ∑_{s=1}^{m+n} t_{is}(u)t′_{sl}(v) − δ_{il} ∑_{s=1}^{m+n} t′_{ks}(v)t_{sj}(u)).  (2)

2 Gauss Decomposition of T (u)

In [1], the Drinfeld presentation is described in terms of the quasideterminants of Gelfand and Retakh ([3], [4]). In this article we make use of the analogous set of generators of the Yangian Y (𝔤𝔩_{m|n}). First we recall the definition of the quasideterminants and some conventional notation.

Definition 2.1. Let X be a square matrix over a ring with identity such that its inverse matrix X^{−1} exists, and such that its jith entry is an invertible element of the ring. Then the ijth quasideterminant of X is defined by the formula

|X|_{ij} = ((X^{−1})_{ji})^{−1} .

Equivalently, we may define quasideterminants inductively as follows. If X = (x_{11}) is a 1 × 1-matrix then there is only one quasideterminant of X; and this is |X|_{11} = x_{11}. For n > 1, we have

|X|_{ij} = x_{ij} − ∑_{k≠i,l≠j} x_{ik}(|X^{ij}|_{lk})^{−1}x_{lj},

where X^{ij} is the matrix obtained from X by removing both the ith row and the jth column. It is sometimes convenient to adopt the following alternative notation for the quasideterminants:

|X|_{ij} =: | \begin{matrix}x_{11} & ⋯ & x_{1j} & ⋯ & x_{1n} \\ & ⋯ & & ⋯ \\ x_{i1} & ⋯ & x_{ij} & ⋯ & x_{in} \\ & ⋯ & & ⋯ \\ x_{n1} & ⋯ & x_{nj} & ⋯ & x_{nn} \end{matrix} | .

The matrix T (u) defined in (1) has the following Gauss decomposition in terms of quasideterminants (by Theorem 4.96 in [3]; see §5 in [1]):

T (u) = F (u)D(u)E(u)

for unique matrices

D(u) = ( \begin{matrix}d_{1}(u) & & ⋯ & 0 \\ & d_{2}(u) & & ⋮ \\ ⋮ & & ... \\ 0 & ⋯ & & d_{m+n}(u) \end{matrix} ) ,

E(u) = ( \begin{matrix}1 & e_{12}(u) & ⋯ & e_{1,m+n}(u) \\ & ... & & e_{2,m+n}(u) \\ & & ... & ⋮ \\ 0 & & & 1 \end{matrix} ) ,

F (u) = ( \begin{matrix}1 & & ⋯ & 0 \\ f_{21}(u) & ... & & ⋮ \\ ⋮ & & ... \\ f_{m+n,1}(u) & f_{m+n,2}(u) & ⋯ & 1 \end{matrix} ) ,

where

d_{i}(u) = | \begin{matrix}t_{11}(u) & ⋯ & t_{1,i−1}(u) & t_{1i}(u) \\ ⋮ & ... & & ⋮ \\ t_{i1}(u) & ⋯ & t_{i,i−1}(u) & t_{ii}(u) \end{matrix} | ,

e_{ij}(u) = d_{i}(u)^{−1} | \begin{matrix}t_{11}(u) & ⋯ & t_{1,i−1}(u) & t_{1j}(u) \\ ⋮ & ... & ⋮ & ⋮ \\ t_{i−1,i}(u) & ⋯ & t_{i−1,i−1}(u) & t_{i−1,j}(u) \\ t_{i1}(u) & ⋯ & t_{i,i−1}(u) & t_{ij}(u) \end{matrix} | ,

f_{ji}(u) = | \begin{matrix}t_{11}(u) & ⋯ & t_{1,i−1}(u) & t_{1i}(u) \\ ⋮ & ... & ⋮ & ⋮ \\ t_{i−1,1}(u) & ⋯ & t_{i−1,i−1}(u) & t_{i−1,i}(u) \\ t_{ji}(u) & ⋯ & t_{j,i−1}(u) & t_{ji}(u) \end{matrix} | d_{i}(u)^{−1}.

It is easy to recover each generating series t_{ij}(u) by multiplying together and taking commutators of d_{i}(u); 1 ≤ i ≤ m+n, and e_{i}(u) := e_{i,i+1}(u), f_{i}(u) = f_{i+1,i}(u); 1 ≤ i < m+n (see §5 of [1]). Thus the Yangian Y (𝔤𝔩_{m|n}) is generated by the coefficients of the latter.

2.1 Some Useful Maps

Here we define some automorphisms of the Yangian Y (𝔤𝔩_{m|n}) and homomorphisms between Yangians, so that we may refer to them in the next section.

Let ω_{m|n} : Y (𝔤𝔩_{m|n}) → Y (𝔤𝔩_{m|n}) be the automorphism defined by
ω : T (u) ↦ T (−u)^{−1}.
Let τ : Y (𝔤𝔩_{m|n}) → Y (𝔤𝔩_{m|n}) be the automorphism defined by
τ(t_{ij}(u)) = t_{ji}(−u) × (−1)^{\overline{i}(\overline{j}+1)}.
Let ρ_{m|n} : Y (𝔤𝔩_{m|n}) → Y (𝔤𝔩_{n|m}) be the isomorphism defined by
ρ_{m|n}(t_{ij}(u)) = t_{m+n+1−i,m+n+1−j}(−u).

Let φ_{m|n} : Y (𝔤𝔩_{m|n}) ↪ Y (𝔤𝔩_{m+k|n}) be the inclusion which sends each generator t_{ij}^{(r)} ∈ Y (𝔤𝔩_{m|n}) to the generator t_{k+i,k+j}^{(r)} in Y (𝔤𝔩_{m+k|n}).

Finally, let ψ_{k} : Y (𝔤𝔩_{m|n}) → Y (𝔤𝔩_{m+k|n}) be the injective homomorphism defined by

ψ_{k} = ω_{m+k|n} ∘ φ_{m|n} ∘ ω_{m|n}.  (3)

This last homomorphism is useful for studying quasideterminants so we discuss it in some detail with the following remarks.

Remark 2.1. We can calculate ψ_{k}(t_{ij}(u)) explicitly for any 1 ≤ i, j ≤ m + n (see Lemma 4.2 of [1]) :

ψ_{k}(t_{ij}(u)) = | \begin{matrix}t_{11}(u) & ⋯ & t_{1k}(u) & t_{1,k+j}(u) \\ ⋮ & ... & ⋮ & ⋮ \\ t_{k1}(u) & ⋯ & t_{kk}(u) & t_{k,k+j}(u) \\ t_{k+i,1}(u) & ⋯ & t_{k+i,k}(u) & t_{k+i,k+j}(u) \end{matrix} | .

In particular, this means that for k ≥ 1, we have ψ_{k}(d_{1}(u)) = d_{k+1}(u), ψ_{k}(e_{1}(u)) = e_{k+1}(u), and ψ_{k}(f_{1}(u)) = f_{k+1}(u).

Furthermore, by (3), we have for any k, l ≥ 1 that ψ_{k} ∘ ψ_{l} = ψ_{k+l}, so we may generalise this observation to give for instance ψ_{k}(d_{l}(u)) = d_{k+l}(u).

Remark 2.2. Notice that the map ψ_{k} sends t_{ij}^{′ (r)} ∈ Y (𝔤𝔩_{m|n}) to the element t_{k+i,k+j}^{′ (r)} in Y (𝔤𝔩_{m+k|n}). Thus the subalgebra ψ_{k}(Y (𝔤𝔩_{m|n})) is generated by the ele-

ments {t_{k+s,k+t}^{′ (r)}}_{s,t=1}^{n}. Then, by (2), all elements of this subalgebra commute with

those of the subalgebra generated by the elements {t_{ij}^{(r)}}_{i,j=1}^{k}.

By Remark 2.1, this implies in particular that for any i, j ≥ 1, the quasideterminants d_{i}(u) and d_{j}(v) commute.

3 The Quantum Berezinian

The quantum Berezinian was defined by Nazarov [7] and plays a similar role in the study of the Yangian Y (𝔤𝔩_{m|n}) as the quantum determinant does in the case of the Yangian Y (𝔤𝔩_{n}) (see [5]).

Definition 3.1. The quantum Berezinian is the following power series with coefficients in the Yangian Y (𝔤𝔩_{m|n}):

b_{m|n}(u) := ∑_{ρ∈S_{m}} sgn(τ) t_{τ(1)1}(u)t_{τ(2)2}(u − 1) ⋯ t_{τ(m)m}(u − m + 1)

× ∑_{σ∈S_{n}}sgn(σ) t′_{m+1,m+σ(1)}(u − m + 1) ⋯ t′_{m+n,m+σ(n)}(u − m + n)

The first part of this expression for b_{m|n}(u) is quite special and so is given its own notation:

C_{m}(u) := ∑_{τ∈S_{m}} sgn(τ)t_{τ(1)1}(u)t_{τ(2)2}(u − 1) ⋯ t_{τ(m)m}(u − m + 1).

It is clear that C_{m}(u) is an element of the subalgebra of Y (𝔤𝔩_{m|n}) generated by the set {t_{ij}^{(r)}}_{1≤i,j≤m;r≥0}. This subalgebra is isomorphic to the Yangian Y (𝔤𝔩_{𝔪}) associated to the Lie algebra 𝔤𝔩_{m} by the inclusion Y (𝔤𝔩_{m}) → Y (𝔤𝔩_{m|n}) which send each generator t_{ij}^{(r)} in Y (𝔤𝔩_{𝔪}) to the generator of the same name in Y (𝔤𝔩_{m|n}). Moreover, C_{m}(u) is in fact the image under this map of the quantum determinant of the smaller Yangian Y (𝔤𝔩_{𝔪}) (see [1], [5]). Then it is well known (see Theorem 2.32 in [6]) that we have the alternative expression:

C_{m}(u) = d_{1}(u)d_{2}(u − 1) ⋯ d_{m}(u − m + 1).

We can extend this observation as follows:

Theorem 1. We have the following alternative expression for the quantum Berezinian:

b_{m|n}(u) = d_{1}(u) d_{2}(u − 1) ⋯ d_{m}(u − m + 1)
× d_{m+1}(u − m + 1)^{−1} ⋯ d_{m+n}(u − m + n)^{−1}.

Proof. Notice that the second part of the expression for b_{m|n}(u) in Definition 3.1 is the image under the isomorphism ρ_{n|m} ∘ ω_{n|m} : Y (𝔤𝔩_{n|m}) → Y (𝔤𝔩_{m|n}) of

∑_{σ∈S_{n}}sgn(σ) t_{n,σ(n)}(u − m + 1) ⋯ t_{2,σ(2)}(u − m + n − 1) t_{1,σ(1)}(u + m − n)  (4)

where in this expression (4) we are following the usual convention for denoting generators in the Yangian Y (𝔤𝔩_{n|m}). We recognise (by comparing with (8.3) of [1] for example) that the expression (4) is in fact C_{n}(u−m+n), the image of the quantum determinant of Y (𝔤𝔩_{n}) under the natural inclusion Y (𝔤𝔩_{n}) ↪ Y (𝔤𝔩_{n|m}). So in order to verify the claim we must calculate the image of C_{n}(u − m + n) under this map explicitly in terms of our quasideterminants d_{i}(v). Applying Proposition 1.6 of [4], we find that the image of d_{i}(v) in Y (𝔤𝔩_{n|m}) is (d_{m+n+1−i}(v))^{−1} in Y (𝔤𝔩_{m|n}). This gives the desired result.

The following theorem is a result of Nazarov [7]. We give a new proof.

Theorem 2. The coefficients of the quantum Berezinian (3.1) are central in the algebra Y (𝔤𝔩_{m|n}).

Proof. By Remark 2.2, we already know that the quantum Berezinian b_{m|n}(u) commutes with d_{i}(v) for 1 ≤ i ≤ m + n. In addition, if we know that the quantum Berezinian commutes with e_{i}(v), then by applying the automorphism τ, we find that it also commutes with f_{i}(−v). So our problem reduces to showing that b_{m|n}(u) commutes with e_{i}(v) for each i between 1 and m + n − 1. We proceed by breaking this problem into three cases.

Case 1: 1 ≤ i ≤ m − 1. For 1 ≤ i ≤ m − 1, we have that e_{i}(v) commutes with C_{m}(u) = d_{1}(u) ⋯ d_{m}(u − m + 1) by Theorem 7.2 in [1]. On the other hand, e_{i}(v) is an element of the subalgebra generated by {t_{jk}^{(r)}}_{1≤j,k≤m} and thus by Remark 2.2 commutes with d_{m+s}(u − m + s)^{−1} = t′_{m+s,m+s}(u − m + s) for 1 ≤ s ≤ n.

Case 2: m + 1 ≤ i ≤ m + n − 1. Applying Propositions 1.6 and 1.4 of [4] in turn to f_{i}(v), we find an alternative expression:

f_{i}(v) = − | \begin{matrix}t′_{i+1,i+1}(v) & ⋯ & t′_{i+1,m+n}(v) \\ ⋮ & & ⋮ \\ t′_{m+n,i+1}(v) & ⋯ & t′_{m+n,m+n}(v) \end{matrix} |^{−1}

· | \begin{matrix}t′_{i+1,i}(v) & t′_{i+1,i+2}(v) & ⋯ & t′_{i+1,m+n}(v) \\ t′_{i+2,i}(v) & t′_{i+2,i+2}(v) & ⋯ \\ ⋮ & ⋮ & ... & ⋮ \\ t′_{m+n,i}(v) & t′_{m+n,i+2}(v) & ⋯ & t′_{m+n,m+n}(v) \end{matrix} | .

Thus, we find that for m + 1 ≤ i ≤ m + n − 1,

e_{i}(v) = ρ_{n|m} ∘ ω_{n|m}(−f_{m+n−i}(v)).

We apply this isomorphism to the results of Case 1 in the Yangian Y (𝔤𝔩_{n|m}). This shows that e_{i}(v) commutes with the quantum Berezinian in the case where m + 1 ≤ i ≤ m + n − 1.

Case 3: i=m. We begin by considering the Yangian Y (𝔤𝔩_{1|1}). For this algebra we have b_{1|1}(u) = d_{1}(u)d_{2}(u)^{−1} and we would like to show that this commutes with e_{1}(v). So it will suffice to show

d_{1}(u)e_{1}(v)d_{2}(u) = d_{2}(u)e_{1}(v)d_{1}(u).  (5)

We have

( \begin{matrix}t_{11}(u) & t_{12}(u) \\ t_{21}(u) & t_{22}(u) \end{matrix} ) = ( \begin{matrix}d_{1}(u) & d_{1}(u) e_{1}(u) \\ f_{1}(u)d_{1}(u) & f_{1}(u)d_{1}(u)e_{1}(u) + d_{2}(u) \end{matrix} )  (6)
( \begin{matrix}t′_{11}(v) & t′_{12}(v) \\ t′_{21}(v) & t′_{22}(v) \end{matrix} ) = ( \begin{matrix}d_{1}(v)^{−1}+ e_{1}(v)d_{2}(v)^{−1}f_{1}(v) & −e_{1}(v) d_{2}(v)^{−1} \\ −d_{2}(v)^{−1}f_{1}(v) & d_{2}(v)^{−1} \end{matrix}) . (7)

An application of (2) gives

(u − v)[t_{11}(u), t′_{12}(v)] = t_{11}(u)t′_{12}(v) + t_{12}(u)t′_{22}(v).

Substituting in the expressions from (6) and (7) then cancelling d_{2}(v), this gives

(u − v)[d_{1}(u), e_{1}(v)] = d_{1}(u)(e_{1}(v) − e_{1}(u)).

Similarly, by considering the commutator [t_{12}(u), t′_{22}(v)], we derive the relation

(u − v)[d_{2}(u), e_{1}(v)] = d_{2}(u)(e_{1}(v) − e_{1}(u)).

We rewrite these relations to find

(u − v)e_{1}(v)d_{1}(u) = (u − v − 1)d_{1}(u)e_{1}(v) + d_{1}(u)e_{1}(u),
(u − v)e_{1}(v)d_{2}(u) = (u − v − 1)d_{2}(u)e_{1}(v) + d_{2}(u)e_{1}(u),

and by considering these expressions we see that (5) holds.

Now we return our attention to the general Yangian Y (𝔤𝔩_{m|n}). By similar appeals to Remark 2.2 as in the first case, we see that e_{m}(v) commutes with d_{1}(u) ⋯ d_{m−1}(u − m + 2) and with d_{m+2}(u − m + 2)^{−1} ⋯ d_{m+n}(u − m + n)^{−1}. So we need only show that e_{m}(v) commutes with d_{m}(u−m+1)d_{m+1}(u−m+1)^{−1}. This follows immediately when we apply the homomorphism ψ_{m−1} to the identity (5) in Y (𝔤𝔩_{1|1}).

References

[1] Jonathan Brundan and Alexander Kleshchev. Parabolic presentations of the Yangian Y (gl_{n}). To appear in Commun.Math. Phys., 2004. Preprint: math.QA/0407011.

[2] Vyjayanthi Chari and Andrew Pressley. A guide to quantum groups. Cambridge University Press, Cambridge, 1994.

[3] I. Gelfand, S. Gelfand, V. Retakh, and R.L. Wilson. Quasideterminants. to appear in Advances in Mathematics, 2004. Preprint: arXiv:math.QA/0208146.

[4] I. M. Gelfand and V. S. Retakh. Determinants of matrices over noncommutative rings. Funct. Anal. Appl., 25(2):91–102, 1991.

[5] A. Molev, M. Nazarov, and G. Olshanskii. Yangians and classical Lie algebras. Russian Math. Surveys, 51:205–282, 1996.

[6] A. I. Molev. Yangians and their applications. In Handbook of algebra, Vol. 3, pages 907–959. North-Holland, Amsterdam, 2003.

[7] M. L. Nazarov. Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys., 21(2):123–131, 1991.