Quasideterminants and Casimir elements for the general Lie superalgebra

arXiv:math/0309461v2 [math.QA] 7 Oct 2003 Quasideterminants and Casimir elements for the general linear Lie superalgebra Alexander Molev and Vladimir Retakh Abstract We apply the techniques of quasideterminants to construct new families of Casimir elements for the general linear Lie superalgebra gl(m|n) whose images under the Harish-Chandra isomorphism are respectively the elementary, complete and power sums supersymmetric functions. School of Mathematics and Statistics University of Sydney, NSW 2006, Australia alexm@maths.usyd.edu.au Department of Mathematics Rutgers University, Piscataway, NJ 08854, USA vretakh@math.rutgers.edu 1 1 Introduction Let A be a square matrix over a ring. Its quasideterminants are certain rational expressions in the entries of A. The theory of quasideterminants originates from the papers by Gelfand and Retakh [2, 3] and since then a number of applications of the theory has been found; see [4] for an overview. In particular, the techniques of quasideterminants is fundamental in the theory of noncommutative symmetric functions developed by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon [1]. The symmetric functions associated with a matrix whose entries are elements of a noncommutative ring is one of the interesting specializations of the general theory. When applied to the matrix E formed by the generators of the general linear Lie algebra gl(n) the theory produces a new family of Casimir elements for gl(n) as well as a distinguished set of generators of the Gelfand–Tsetlin subalgebra of U(gl(n)); see [1, Section 7.4]. These results were extended to the orthogonal and symplectic Lie algebras in [8] with the use of the twisted Yangians and quantum determinants; see also a review paper [10]. In this paper we use the techniques of quasideterminants to get new families of Casimir elements for the general linear Lie superalgebra gl(m|n) and calculate their images with respect to the Harish-Chandra isomorphism. They can be regarded as super-analogs of those constructed in [1, Section 7.4]. Three families of Casimir elements are given explicitly in terms of some oriented graphs associated with gl(m|n). The Harish-Chandra images turn out to be respectively the elementary, complete and power sums supersymmetric functions. The starting point for our construction is a result of Nazarov [12]. He produced a formal series B(t) called quantum Berezinian with coefficients in the center of the universal enveloping algebra U(gl(m|n)). Our first result is a quasideterminant factorization of B(t) (Theorem 3.1). We then use it to get graph presentations for the Casimir elements (Theorem 4.1). Some other families of Casimir elements for gl(m|n) were constructed e.g. in [9]. This work is a super-version of the earlier constructions of [13, 14] for gl(n) and it provides a linear basis of the center of U(gl(m|n)) formed by the so-called quantum immanants. We acknowledge the financial support of the Australian Research Council. The second author would like to thank the School of Mathematics and Statistics, the University of Sydney, for the warm hospitality during his visit. His work was also partially supported by NSA. 2 2 Preliminaries Let x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ) be two families of variables. A polynomial P in x and y is called supersymmetric if P is symmetric separately in x and y and satisfies the following cancellation property: the result of setting xm = −yn = z in P is independent of z. We denote by Λ(m|n) the algebra of supersymmetric polynomials in x and y. The algebra Λ(m|n) is generated by the polynomials pk = xk1 + · · · + xkm + (−1)k−1 (y1k + · · · + ynk ), k ≥ 1, (2.1) called the power sums supersymmetric functions. Two other families of generators of Λ(m|n) are comprised by the elementary and complete supersymmetric functions defined respectively by the formulas X X X xi1 · · · xip yj1 · · · yjq , ek = p+q=k i1 <···<ip j1 ≤···≤jq hk = X X X xi1 · · · xip yj1 · · · yjq ; (2.2) p+q=k i1 ≤···≤ip j1 <···<jq see [15], [16]. We shall denote by Eij , i, j = 1, . . . , m + n the standard basis of the Lie superalgebra gl(m|n). The Z2 -grading on gl(m|n) is defined by Eij 7→ ı̄ + ̄, where ı̄ is an element of Z2 which equals 0 or 1 depending on whether i ≤ m or i > m. The commutation relations in this basis are given by [Eij , Ekl ] = δkj Eil − δil Ekj (−1)(ı̄+̄)(k̄+l̄) . (2.3) Given a m+n-tuple (λ|µ) = (λ1 , . . . , λm , µ1 , . . . , µn ) ∈ C m+n we consider a highest weight gl(m|n)-module L(λ|µ) with the highest weight (λ|µ). That is, L(λ|µ) is generated by a nonzero vector ξ such that Eii ξ = λi ξ for i = 1, . . . , m, Em+j,m+j ξ = µj ξ for j = 1, . . . , n, Eij ξ = 0 (2.4) for 1 ≤ i < j ≤ m + n. Any element z of the center Z(gl(m|n)) of the universal enveloping algebra U(gl(m|n)) acts in L(λ|µ) as a scalar χ(z). For a fixed z the scalar χ(z) is a polynomial in λi and µi which is supersymmetric in the shifted variables defined by xi = λi − i + 1 for i = 1, . . . , m, y j = µj + m − j for j = 1, . . . , n. (2.5) Furthermore, the map z 7→ χ(z) defines an algebra isomorphism χ : Z(gl(m|n)) → Λ(m|n), which is called the Harish-Chandra isomorphism; see [5], [15], [16]. 3 (2.6) 3 Decomposition of the Quantum Berezinian b of size (m + n) × (m + n) whose ij-th entry is E bij = Introduce the super-matrix E ̄ (−1) Eij . By the quantum Berezinian we mean the formal series B(t) defined by X

b b − m + 1) sgn σ 1 + t E · · · 1 + t ( E B(t) = σ(1),1 σ(m),m σ∈Sm × X τ ∈Sn b − m + 1) sgn τ 1 + t (E
−1 m+1,m+τ (1) b − m + n) · · · 1 + t (E
−1 m+n,m+τ (n) . (3.1) The quantum Berezinian was constructed by Nazarov [12]. He also proved that all its coefficients are central in the universal enveloping algebra U(gl(m|n)). The image of B(t) under the Harish-Chandra isomorphism is given by
(1 + tx1 ) · · · (1 + txm ) χ B(t) = , (1 − ty1 ) · · · (1 − tyn ) (3.2) cf. [9]. Our first result is a decomposition of B(t) into a product of quasideterminants. If X is a square matrix over a ring with 1 such that there exists the inverse matrix X −1 and its ji-th entry (X −1 )ji is an invertible element of the ring, then the ij-th quasideterminant of X is defined by the formula |X|ij = (X −1 )ji
−1 , see [2, 3] for other equivalent definitions of the quasideterminants and their properties. Theorem 3.1. We have the following decomposition of B(t) in the algebra of formal series with coefficients in U(gl(m|n)) b (1) B(t) = 1 + t E 11 b(m) − m + 1) · · · 1 + t (E b (m+1) − m + 1) × 1 + t (E −1 m+1,m+1 mm b(m+n) − m + n) · · · 1 + t (E −1 , m+n,m+n (3.3) b(k) denotes the submatrix of E b corresponding to the first k rows and columns. where E Moreover, the factors in the decomposition are pairwise permutable. Proof. We employ a quasideterminant decomposition of the quantum determinant for the Yangian Y(gl(r)). The latter is the associative algebra with the generators (1) (2) tij , tij , . . . where 1 ≤ i, j ≤ r and the following defining relations [tij (u), tkl (v)] =
1 tkj (u)til (v) − tkj (v)til (u) , u−v 4 (3.4) where (1) (2) tij (u) = δij + tij u−1 + tij u−2 + · · · ∈ Y(gl(n))[[u−1 ]]. (3.5)   Consider the quantum determinant of the matrix T (u) = tij (u) defined by the following equivalent formulas X sgn σ · tσ(1),1 (u) · · · tσ(r),r (u − r + 1) qdet T (u) = σ∈Sr = X (3.6) sgn σ · t1,σ(1) (u − r + 1) · · · tr,σ(r) (u). σ∈Sr It is well-known that the coefficients of this series are algebraically independent generators of the center of the algebra Y(gl(r)); see e.g. [11] for a proof. For 1 ≤ k ≤ n denote by T (k) (u) the submatrix of T (u) corresponding the first k rows and columns. We have the following quasideterminant decomposition of qdet T (u) in the algebra Y(gl(m))[[u−1 ]] qdet T (u) = |T (1) (u)|11 · · · |T (m) (u − m + 1)|mm , (3.7) where the factors are pairwise permutable; see [8] and also [2], [6] for analogous decompositions in the case of noncommutative determinants of different types. Now we apply the algebra homomorphism Y(gl(m)) → U(gl(m|n)) given by b (m) u−1 T (u) 7→ 1 + E (3.8) qdet T (u) = |T(1) (u − n + 1)|11 · · · |T(n) (u)|nn . (3.9) to (3.7), set u = t−1 and multiply both sides by (1 − t) · · · (1 − (m − 1)t). This will represent the first determinant factor in (3.1) as a product of quasideterminants which comprise the first m factors in (3.3); cf. [8]. Now consider the second factor in (3.1). We shall use the subscript (k) of a matrix to indicate its submatrix obtained by removing the first k −1 rows and columns. Here we need another version of the decomposition (3.7) given by Apply another homomorphism Y(gl(n)) → U(gl(m|n)) defined by   b u−1 )−1 T (u) 7→ (1 + E , (m+1) (3.10) (see [12]) to both sides of (3.9) with qdet T (u) expanded by the second formula in (3.6). Now observe that by the Inversion Theorem for quasiminors [2, 3], we have for any k ∈ {1, . . . , n}  b (u − n + k)−1)−1 (1 + E  (m+k) m+k,m+k b (m+k) (u − n + k)−1 = 1+E 5 −1 . m+k,m+k (3.11) To complete the argument, it remains to set u = t−1 + n − m and divide both sides of the relation by the product (1 + t(1 − m)) · · · (1 + t(n − m)). Finally, note that the product of the first m + n − 1 factors in (3.3) coincides with the quantum Berezinian for the subalgebra gl(m|n − 1) of gl(m|n). Therefore the last factor in (3.3) is permutable with the elements of gl(m|n − 1) by the centrality of the quantum Berezinian. The proof is completed by an obvious induction. 4 Casimir elements Let A = (Aij ) be a square matrix of size l × l with entries from an arbitrary ring and let t be a formal variable. Fix an integer i between 1 and l. Following [1, Definition 7.19] introduce the noncommutative symmetric functions associated with (i) the matrix A and the index i as follows. The elementary symmetric functions Λk , (i) the complete symmetric functions Sk , the power sums symmetric functions of the (i) (i) first kind Ψk and the power sums symmetric functions of the second kind Φk are defined by the formulas 1+ ∞ X Λk tk = |1 + tA|ii , 1+ ∞ X Sk tk = |1 − tA|−1 ii , (i) k=1 (i) k=1 ∞ X k=1 ∞ X k=1 (i) Ψk tk−1 (i) d = |1 − tA|ii |1 − tA|−1 ii , dt Φk tk−1 = − (4.1)
d log |1 − tA|ii . dt These functions are polynomials in the entries of the matrix A and can be interpreted in terms of graphs in the following way. Let us consider the complete oriented graph A with l vertices {1, 2, . . . , l}, the arrow from i to j being labelled by Aij . Then every path in the graph going from i to j defines a monomial of the form Air1 Ar1 r2 · · · Ark−1 j . A simple path is a path such that rs 6= i, j for every s. Then by [1, Proposition 7.20], (i) (−1)k−1 Λk is the sum of all monomials labelling simple paths in A of length k going (i) from i to i; Sk is the sum of all monomials labelling paths in A of length k going (i) from i to i; Ψk is the sum of all monomials labelling paths in A of length k going from i to i, where the coefficient of each monomial is the length of the first return to (i) i; Φk is the sum of all monomials labelling paths in A of length k going from i to i, where the coefficient of each monomial is the ratio of k to the number of returns to i. 6 b (i) − i + 1 and the noncommutative For any i = 1, . . . , m consider the matrix E symmetric functions associated with this matrix and the index i. We keep the above notation for these functions. Similarly, for any j = 1, . . . , n consider the matrix b(m+j) + m − j and the noncommutative symmetric functions associated with this −E matrix and the index m + j. Again, we denote the functions by the same symbols and distinguish them by the upper index m + j. Theorem 4.1. The algebra Z(gl(m|n)) is generated by each of the families X (m+n) (1) (m) (m+1) Λk = Λi1 · · · Λim Sim+1 · · · Sim+n , i1 +···+im+n =k Sk = X (1) (m) (m+1) (m+n) Si1 · · · Sim Λim+1 · · · Λim+n , i1 +···+im+n =k m n X X (i) (m+j) k−1 Ψk + (−1) Ψk = Ψk , i=1 j=1 m n X X (i) (m+j) k−1 Φk + (−1) Φk , Φk = i=1 j=1 (4.2) where k = 1, 2, . . . . Moreover, Ψk = Φk for any k, and the Harish-Chandra images of these generators are respectively the elementary, complete and power sums supersymmetric functions, χ(Λk ) = ek , χ(Sk ) = hk , χ(Ψk ) = pk . (4.3) Proof. Introduce the generating functions for the supersymmetric polynomials (2.1) and (2.2) by ∞ X p(t) = pk tk−1 , k=1 e(t) = 1 + h(t) = 1 + ∞ X k=1 ∞ X ek tk , (4.4) hk tk . k=1 These functions are related by h(t) = e(−t)−1 , p(t) = − d d log e(−t) = e(−t) e(−t)−1 , dt dt (4.5) see e.g. [7]. On the other hand, by Theorem 3.1 we have 1+ ∞ X Λk tk = B(t) k=1 7 (4.6) which proves that the elements Λk are central in U(gl(m|n)). Moreover, χ(B(t)) = e(t) due to (3.2) and so χ(Λk ) = ek . The proof is completed by applying (4.5) and taking into account the fact that the factors in the decomposition (3.3) are mutually permutable; cf. the argument for the case of gl(n) [1, Section 7.4]. Example 4.2. We have Ψ1 = m X i=1 Ψ2 = n X (Eii − i + 1) + (Em+j,m+j + m − j), 2 (Eii − i + 1) + 2 i=1 − j=1 m  X n  X i−1 X Eik Eki  (4.7) k=1  (−1) Em+j,l El,m+j . m+j−1 2 (Em+j,m+j + m − j) − 2 j=1 X l̄ l=1 References [1] I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), 218–348. [2] I. M. Gelfand and V. S. Retakh, Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 (1991), 91-102. [3] I. M. Gelfand and V. S. Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992), 1-20; Publ. LACIM, UQAM, Montreal, 14, 1-26. [4] I. Gelfand, S. Gelfand, V. Retakh and R. Wilson, Quasideterminants, preprint math.QA/0208146. [5] V. G. Kac, Representations of classical Lie superalgebras, in: “Differential Geometry Methods in Mathematical Physics II”, (K. Bleuer, H. R. Petry, A. Reetz, Eds.), Lecture Notes in Math., Vol. 676, pp. 597–626. Springer-Verlag, Berlin/Heidelberg/New York, 1978. [6] D. Krob and B. Leclerc, Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995), 1–23. [7] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 2nd edition 1995. 8 [8] A. I. Molev, Noncommutative symmetric functions and Laplace operators for classical Lie algebras, Lett. Math. Phys. 35 (1995), 135-143. [9] A. I. Molev, Factorial supersymmetric Schur functions and super Capelli identities, in: “Kirillov’s Seminar on Representation Theory”, (G. I. Olshanski, Ed.), Amer. Math. Soc. Transl., Ser. 2, Vol. 181, AMS, Providence, R.I., 1998, pp. 109–137. [10] A. I. Molev, Yangians and their applications, in: “Handbook of Algebra”, Vol. 3, (M. Hazewinkel, Ed.), Elsevier, 2003. [11] A. Molev, M. Nazarov and G. Olshanski, Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), 205–282. [12] M. L. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991), 123–131. [13] M. L. Nazarov, Yangians and Capelli identities, in: “Kirillov’s Seminar on Representation Theory”, (G. I. Olshanski, Ed.), Amer. Math. Soc. Transl., Ser. 2, Vol. 181, AMS, Providence, R.I., 1998, pp. 139–163. [14] A. Okounkov, Quantum immanants and higher Capelli identities, Transformation Groups 1 (1996), 99–126. [15] M. Scheunert, Casimir elements of Lie superalgebras, in: “Differential Geometry Methods in Mathematical Physics”, pp. 115–124, Reidel, Dordrecht, 1984. [16] J. R. Stembridge, A characterization of supersymmetric polynomials, J. Algebra 95 (1985), 439–444. 9