Related papers: Note on dilogarithm identities from nilpotent doub…

We define the degenerate two boundary affine Hecke-Clifford algebra $\mathcal{H}_d$, and show it admits a well-defined $\mathfrak{q}(n)$-linear action on the tensor space $M\otimes N\otimes V^{\otimes d}$, where $V$ is the natural module…

We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is…

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite…

Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the…

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the…

Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare…

We describe a bilinear identity satisfied by certain multidimensional q-hypergeometric integrals. The identity can be considered as a deformation of the Riemann bilinear relation for the twisted de Rham (co)homologies. The identity also…

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…

In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0,…

This paper provides the first algebraic characterization of an algebra of cohomological Hecke operators associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular…

We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in…

Let $\mathfrak{g}$ be a simple finite dimensional complex Lie algebra and let $\widehat{\mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a…

Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is…

In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to…

To any Frobenius superalgebra $A$ we associate towers of Frobenius nilCoxeter algebras and Frobenius nilHecke algebras. These act naturally, via Frobenius divided difference operators, on Frobenius polynomial algebras. When $A$ is the…

We introduce a class of induced representations of the degenerate double affine Hecke algebra of gl_N and analyze their structure mainly by means of intertwiners. We also construct them from modules of the affine Lie algebra using…

In this paper, we prove a new Rogers-Ramanujan-type identity, involving grounded partitions, by computing a character of the affine Kac-Moody algebra $D_4^{(3)}$ in two different ways. The product side is derived using Lepowsky's product…

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, $\mathscr{C}_{\mathscr{Q}}^{(1)}$ and…

In the paper, we introduce and calculate difference Fourier transforms on representations of the double affine Hecke algebras in polynomilas, polynomials multiplied by the Gaussian, and various spaces of delta-functions including…

The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or…