Related papers: Polynomial functors and two-parameter quantum symm…
Classical A-polynomials $A(\ell,m)$ define constraints on coordinates $\ell$ and $m$ in $SL(2,\mathbb{C})$ (a complexification of $SU(2)$) character varieties associated to knot complements $S^3\setminus K$. Quantum A-polynomials $\hat…
We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau…
We categorify a coideal subalgebra of the quantum group of $\mathfrak{sl}_{2r+1}$ by introducing a $2$-category \`a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this…
Let G=GL(N), K=GL(p)xGL(q), where p+q=N, and n be a positive integer. We construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type…
We study the quantum symmetric spaces for quantum general linear groups modulo symplectic groups. We first determine the structure of the quotient quantum group and completely determine the quantum invariants. We then derive the…
In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend…
We consider some general aspects of the new noncommutative or quantum geometry coming out of the theory of quantum groups, in connection with Planck scale physics. A generalisation of Fourier or wave-particle duality on curved spaces…
Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_qL\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^{\theta}{\sf F}$ between finite-dimensional modules over a quantum symmetric pair of affine type…
We establish a q-version of the Schur-Weyl duality, in which the role of the symmetric group is played by the Hecke algebra and the role of the enveloping algebra U(gl(N)) is played by the Reflection Equation algebra, associated with any…
We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q…
According to the Hall algebras of quivers with automorphisms under Lusztig's construction, the polynominal forms of several structure coefficients for quantum groups of all finite types are presented in this note. We first provide a…
We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies…
We establish a q-generalization of Gordon's theorem that the space of diagonal coinvariants has a quotient identified with a perfect representation of the rational double affine Hecke algebra. It leads to a simple proof of his theorem and…
Quantum analogues of the homogeneous spaces $\GL(n)/\SO(n)$ and $\GL(2n)/\Sp(2n)$ are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials…
A many variable $q$-calculus is introduced using the formalism of braided covector algebras. Its properties when certain of its deformation parameters are roots of unity are discussed in detail, and related to fractional supersymmetry. The…
Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic…
We introduce dual Hopf algebras which simultaneously combine the concepts of the k-Schur function theory with the quasi-symmetric Schur function theory. We construct dual basis of these Hopf algebras with remarkable properties.
We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a set of degree--lowering operators on the space of polynomials in $n$ variables. This allows the construction of raising/lowering operators with a simple action on…
We construct quantum supersymmetric pairs $({\bold U},{\bold U}^\imath)$ of type AIII and elucidate their fundamental properties. An $\imath$Schur duality between the $\imath$quantum supergroup ${\bold U}^\imath$ and the Hecke algebra of…
We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence…