q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra Uq(u(n,1))

For the quantum algebra Uq(gl(n+1)) in its reduction on the subalgebra Uq(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Zq(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra Uq(u(n,1)) which is a real form of Uq(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form

Feigin-Frenkel center in types B, C and D

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) 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 V_{crit}(g) associated with the simple Lie algebras g 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 sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and commutative subalgebras in universal enveloping algebras are adde

Some aspects of q- and qp-Boson calculus

Communication to : Symmetrices in science VII, Brengenz (AU), 1994-08-07SIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : RP 13041 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc