We consider Casimir elements for the orthogonal and symplectic Lie algebras constructed with the use of the Brauer algebra. We calculate the images of these elements under the Harish-Chandra isomorphism and thus show that they (together with the Pfaffian-type element in the even orthogonal case) are algebraically independent generators of the centers of the corresponding universal enveloping algebras. (C) 2013 Elsevier Inc. All rights reserved.

The classical characteristic map associates symmetric functions to characters of the symmetric groups. There are two natural analogues of this map involving the Brauer algebra. The first of them relies on the action of the orthogonal or symplectic group on a space of tensors, while the second is provided by the action of this group on the symmetric algebra of the corresponding Lie algebra. We consider the second characteristic map both in the orthogonal and symplectic case, and calculate the images of central idempotents of the Brauer algebra in terms of the Schur polynomials. The calculation is based on the Okounkov-Olshanski binomial formula for the classical Lie groups. We also reproduce the hook dimension formulas for representations of the classical groups by deriving them from the properties of the primitive idempotents of the symmetric group and the Brauer algebra.

For each simple Lie algebra consider the corresponding affine vertex algebra at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras associated with the simple Lie algebras of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras and , and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra . We also introduce analogues of the Bethe subalgebras of the Yangians and show that their graded images coincide with the respective commutative subalgebras of .

We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables. The function takes values in the Brauer algebra and has the form of a product of R-matrix type factors. In particular, this provides a one-parameter version of the fusion procedure for the symmetric group. The R-matrices are solutions of the Yang-Baxter equation associated with the classical Lie algebras g(N) of types B, C, and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g(N)) to U(g(N)) and show that the fusion procedure provides an equivalence between natural tensor representations of B(g(N)) with the corresponding evaluation modules.

We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu-Markov trace of the idempotents.

We give a new version of the fusion procedure for the symmetric group which originated in the work of Jucys and was developed by Cherednik. We derive it from the Jucys-Murphy formulae for the diagonal matrix units for the symmetric group.

An explicit realization of the skew representations of the quantum affine algebra U-q((gl) over cap (n)) is given. It is used to identify these representations in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (or q-character).

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL(N) and the Yangian for gl(N). We prove a version of this theorem for the twisted Yangians Y(g(N)) associated with the orthogonal and symplectic Lie algebras h(N) = o(N) or sp(N). This gives rise to representations of the twisted Yangian Y(g(N-M)) on the space of homomorphisms Hom(gM) (W, V), where W and V are finite-dimensional irreducible modules over g(M) and g(N), respectively. In the symplectic case these representations turn out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials. We also apply the quantum Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras.

We describe the algebra of invariants of the vacuum module associated with an affinization of the Lie superalgebra gl(1 vertical bar 1). We give a formula for its Hilbert-Poincare series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the (1,1)-hook. Our arguments are based on a super version of the Beilinson-Drinfeld-Rais-Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with gl(1 vertical bar 1). We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.

We construct an action of the braid group B-N on the twisted quantized enveloping algebra U'(q)(o(N)) where the elements of B-N act as automorphisms. In the classical limit q -> 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra U'(q)(sp(2n)). We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

We construct an action of the braid group B_N on the twisted quantized enveloping algebra $\U_{q}^{'}(\mathfrak{o}_{N})$ where the elements of B_N act as automorphisms. In the classical limit q 鈫?1, we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra $\U_{q}^{'}(\mathfrak{sp}_{2n})$. We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras. [ABSTRACT FROM AUTHOR]

We introduce two subalgebras in the type A quantum affine algebra which are coideals with respect to the Hopf algebra structure. In the classical limit q鈫? each subalgebra specializes to the enveloping algebra [formula], where [formula] is a fixed point subalgebra of the loop algebra [formula] with respect to a natural involution corresponding to the embedding of the orthogonal or symplectic Lie algebra into [formula]. We also give an equivalent presentation of these coideal subalgebras in terms of generators and defining relations which have the form of reflection-type equations. We provide evaluation homomorphisms from these algebras to the twisted quantized enveloping algebras introduced earlier by Gavrilik and Klimyk and by Noumi. We also construct an analog of the quantum determinant for each of the algebras and show that its coefficients belong to the center of the algebra. Their images under the evaluation homomorphism provide a family of central elements of the corresponding twisted quantized enveloping algebra. [ABSTRACT FROM AUTHOR]