arXiv subjects

Igor Frenkel

Publications and source records attributed to Igor Frenkel.

30 records · Page 2Linked to original sources

arXiv2003

Quantum instantons with classical moduli spaces

We introduce a quantum Minkowski space-time based on the quantum group SU(2)q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective twistor space.

math.QA↗

arXiv2002

Quiver varieties, affine Lie algebras, algebras of BPS states, and semicanonical basis

We suggest a (conjectural) construction of a basis in the plus part of the affine Lie algebra of type ADE indexed by irreducible components of certain quiver varieties. This construction is closely related to a string-theoretic construction of a Lie algebra of BPS states. We then study the new combinatorial questions about the (classical) root systems naturally arising from our constructions and Lusztig's semicanonical basis.

math-ph↗

arXiv2001

Virasoro algebra and wreath product convolution

We present a group theoretic construction of the Virasoro algebra in the framework of wreath products. This can be regarded as a counterpart of a geometric construction of Lehn in the theory of Hilbert schemes of points on a surface.

math.QA↗

arXiv2000

Twisted vertex representations via spin groups and the McKay correspondence

We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group $Γ$ and a virtual character of $Γ$ we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products $Γ\wr\widetilde{S}_n$ of $Γ$ and a double cover of the symmetric group $S_n$ for all $n$. When $Γ$ is a subgroup of $SL_2(\mathbb C)$ with the McKay virtual character, our construction gives a group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras. When $Γ$ is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for $Γ\wr\widetilde{S}_n$.

math.QA↗

arXiv2000

Affine Lie Algebras and Tame Quivers

C.M. Ringel defined Hall algebra associated with the category of representations of a quiver of Dynkin type and gave an explicit description of the structure constants of the corresponding Lie algebra. We utilize functorial properties of the Hall algebra to give a simple proof of Ringel's result, and to generalize it to the case of a quiver of affine type.

math.RT↗

arXiv2000

A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors

We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the wall functors act as generators of the Temperley-Lieb algebra while Zuckerman functors act as generators of U(sl(2)).

math.QA↗

arXiv1999

Quantum vertex representations via finite groups and the McKay correspondence

We establish a $q$-analog of our recent work on vertex representations and the McKay correspondence. For each finite group $Γ$ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ\times \mathbb C^{\times}$ and the symmetric groups. An important special case is obtained when $Γ$ is a finite subgroup of $SU_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.

math.QA↗

arXiv1999

Vertex representations via finite groups and the McKay correspondence

Given a finite group $Γ$ and a virtual character $\wt$ on it, we construct a Fock space and associated vertex operators in terms of representation ring of wreath products $Γ\sim S_n$. We recover the character tables of wreath products $Γ\sim S_n$ by vertex operator calculus. When $Γ$ is a finite subgroup of $SU_2$, our construction yields a group theoretic realization of the basic representations of the affine and toroidal Lie algebras of $ADE$ type, which can be regarded as a new form of McKay correspondence.

math.QA↗

arXiv1998

Canonical Basis and Macdonald Polynomials

In the basic representation of $U_q(\hat{sl}(2))$ realized via the algebra of symmetric functions we compare the canonical basis with the basis of Macdonald polynomials with $q=t^2$. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also discuss the positivity and integrality of these coefficients. For level $k$, we expect a similar relation between the canonical basis and Macdonald polynomials with $q^2=t^{k}.$

math.QA↗

arXiv1997

Kazhdan-Lusztig polynomials and canonical basis

In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of the quantum group $U_q sl_k$. We also use known results about canonical bases for $U_q sl_2$ to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmanians), due to Lascoux-Schutzenberger and Zelevinsky.

q-alg↗

arXiv1997

Canonical basis and homology of local systems

Using the isomorphism between highest weight U_q(sl_2)-modules and homologies of certain local systems on the configuration spaces, constructed by Varchenko, we give a geometric construction of the dual of the Lusztig's canonical basis in a tensor product of irreducible finite-dimensional U_q(sl_2)-modules.

q-alg↗

arXiv1994

Spherical functions on affine Lie groups

We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.

hep-th↗