LatticeW algebras and quantum groups

We present Feigin's construction [Lectures given in Landau Institute] of latticeW algebras and give some simple results: lattice Virasoro andW ₃ algebras. For the simplest caseg=sl(2), we introduce the wholeU _q(2)) quantum group on this lattice. We find the simplest two-dimensional module as well as the exchange relations and define the lattice Virasoro algebra as the algebra of invariants ofU _q(sl(2)). Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine groupU _q(ñ₊). We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 132–147, July, 1994.     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Strip Bundles and the Highest Weight Crystals Over Quantum Infinite Rank Affine Algebras

We give new realizations of the highest weight crystals B(λ) over the quantum infinite rank affine algebras U _q (A _∞), U _q (B _∞), U _q (C _∞), and U _q (D _∞) using strip bundles, which are related with Nakajima monomials.     

The representations ofU q(SO4) andU q(SO3,1)

Summary We have considered here the (unitary) irreducible representations of theq-deformed algebraU _q(SO₄) and of theq-deformed Lorentz algebraU _q(SO_3,1). Both of them contain, as subalgebra, the algebraU _q(SO₃) which is shown to be isomorphic to the Fairlie-Odesskii algebra. As the list of pairwise nonequivalent irreps of theU _q(SO_3,1) demonstrates, the set of the parameters, which characterize such irreps is somewhat reduced (due to periodicity properties of the function w(z)=[z]_q) in comparison with that of theq=1 (classical) case. From another side, the list of unitary irreps of theU _q(SO_3,1) contains the strange series which has no classical counterpart (disappears at q=1).Published in Teoreticheskaya i Matematicheskaya Fizika. Vol. 95, No. 2, pp. 251–257, May, 1993.     

On the equivalence of plane curve singularities

A systematic method to calculate cleft extensions for pointed Hopf algebras is developed and applied to U_q(sl ₂) and the Frobenius-Lusztig kernel U_q(sl ₂)'.     

Conjugacy Classes in Sylow p-Subgroups of Finite Chevalley Groups in Bad Characteristic

Let U = U(q) be a Sylow p-subgroup of a finite Chevalley group G = G(q). Röhrle and Goodwin in 2009 determined a parameterization of the conjugacy classes of U, for G of small rank when q is a power of a good prime for G. As a consequence they verified that the number k(U) of conjugacy classes of U is given by a polynomial in q with integer coefficients. In the present paper, we consider the case when p is a bad prime for G. Our motivation is to observe how the situation differs between good and bad characteristics. We obtain a parameterization of the conjugacy classes of U, when G has rank less than or equal to 4, and G is not of type F ₄. In these cases we deduce that k(U) is given by a polynomial in q with integer coefficients; this polynomial is different from the polynomial for good primes.     

On the Local Structure of Generalized Elliptic Pseudodifferential Operators and the Gauss Method

In this paper, we study the class of operators whose dominant parts admit elliptic factorizations in a conic domain U from T'(X), i.e., they can be represented as the composition of diagonal operators and operators of zero order, elliptic in U. We denote this class by ELF ^°(U). It arises in the process of microlocalization of the notion of generalized ellipticity. We analyze the problem concerning the simplest factorization of the dominant part of the operator BAC, where A EFL ^°(U) and the operators B and C are chosen from the class EL ^°(U_q)(elliptic operators in a neighborhood U _q of the point q U). For A from the subclass denoted by BEL ^°(U), the dominant part BAC can be reduced to a single diagonal operator. In general, for operators from the class EFL ^°(U) such a representation may not exist. However, there always exists a representation whose dominant part BAC is composed of a finite number of diagonal operators, permutation matrices, and lower triangular matrices having units on the main diagonal. We prove this fact by using an analog of the Gauss method, which we introduce for the algebra of pseudodifferential operators. Bibliography: 5 titles.     

An Algorithm for Computing the Global Basis of a Finite Dimensional Irreducible U q ( so 2 n + 1 ) or U q ( so 2 n )-Module

Abstract

We describe a simple algorithm for computing the canonical basis of any irreducible finite-dimensional U _q (s o _2n+1) or U _q (s o _2n )-module.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

The Structure of Weak Hopf Algebras Corresponding to U q (sl 2)

The structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to U _q (sl ₂) can be written as the direct sum of U _q (sl ₂) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their Ext quiver. There are four types of such structures.     

Schensted-Type Correspondences and Plactic Monoids for Types B n and D n

We use Kashiwara's theory of crystal bases to study plactic monoids for U _q(so _2n+1) and U _q(so _2n ). Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type B from the Sheats sliding algorithm.     

Leonard Pairs and Leonard Triples of q-Racah Type from the Quantum Algebra U q (sl 2)

In this article, we describe the construction of Leonard pairs and Leonard triples that have q-Racah type from U _q (sl ₂)-modules by using equitable generators of U _q (sl ₂). Our result solves an open problem proposed by Terwilliger.     

Quantum groups and quantum shuffles

Let U _q ⁺ be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix. We show that U _q ⁺ is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z ⁿ . This method gives supersymetric as well as multiparametric versions of U _q ⁺ in a uniform way (for a suitable choice of the Hopf bimodule). We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition. We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U _q sl(2) and a suitable irreducible finite dimensional representation. Oblatum 21-III-1997 & 12-IX-1997     

Monomial Realization of the Crystals Over U q (A ∞) and the Quantum Monster Algebra

We give new realizations of the crystal bases of the Verma modules and the irreducible highest weight modules over the quantum generalized Kac–Moody algebra U _q (A _∞) and the quantum Monster algebra using Nakajima monomials. Moreover, another realization of the crystals B(∞) and B(λ) over U _q (A _∞) using triangular matrices and tableaux are given.     

Unipotent Conjugacy in General Linear Groups

Let U(n,q) be the group of upper uni-triangular matrices in GL(n,q), the n-dimensional general linear group over the field of q elements. The number of U(n,q)-conjugacy classes in GL(n,q) is, as a function of q, for fixed n, a polynomial in q with integral coefficients.     

Embeddings of Un(q2) and symmetric strongly regular graphs

In the geometric setting of the embedding of the unitary group U_n(q²) inside an orthogonal or a symplectic group over the subfield GF(q) of GF(q²), q odd, we show the existence of infinite families of transitive two‐character sets with respect to hyperplanes that in turn define new symmetric strongly regular graphs and two‐weight codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 248–253, 2010     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

THE CLASSIFICATION OF CONVEX ORDERS ON AFFINE ROOT SYSTEMS

We classify all total orders with a convex property on the positive root system of an arbitrary untwisted affine Lie algebra g. Such total orders are called convex orders and are used to construct convex bases of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra U_q ⁺ of the quantized universal enveloping algebra U_q (g).     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.