Фейгин Борис Львович

Quantum continuous gl∞: Tensor products of Fock modules and Wn-characters

2011 · ARTICLE · en

Quantum continuous gl∞: semiinfinite construction of representations

2011 · ARTICLE · en

Gelfand-Zetlin Basis, Whittaker Vectors and a Bosonic Formula for the sl(n+1) Principal Subspace

2011 · ARTICLE · ru

Gaudin models with irregular singularities

2010 · ARTICLE · en

Opers with irregular singularity and spectra of the shift of argument subalgebra

2010 · ARTICLE · en

"The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380 , math.QA/0612798 . We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian."

A commutative algebra on degenerate CP1 and Macdonald polynomials

2009 · ARTICLE · en

We introduce a unital associative algebra associated with degenerate CP1. We show that is a commutative algebra and whose Poincare' series is given by the number of partitions. Thereby, we can regard as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding-Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard-Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov-Pandharipande [e-print arXiv:math-ph/0411210].

Fermionic Formulas for Eigenfunctions of the Difference Toda Hamiltonian

2009 · ARTICLE · en

We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.

Lusztig limit of quantum sℓ(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models

2009 · ARTICLE · en

We introduce a Kazhdan–Lusztig-dual quantum group for (1,p) Virasoro logarithmic minimal models as the Lusztig limit of the quantum sℓ(2) at pth root of unity and show that this limit is a Hopf algebra. We calculate tensor products of irreducible and projective representations of the quantum group and show that these tensor products coincide with the fusion of irreducible and logarithmic modules in the (1,p) Virasoro logarithmic minimal models.

Form factors of descendant operators: free field construction and reflection relations

2009 · ARTICLE · en

On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level

2009 · ARTICLE · en