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We define global and local Weyl modules for Lie superalgebras of the form $\mathfrak{g} \otimes A$, where $A$ is an associative commutative unital $\mathbb{C}$-algebra and $\mathfrak{g}$ is a basic Lie superalgebra or $\mathfrak{sl}(n,n)$,…

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 use duality theorems to obtain presentations of some categories of modules. To derive these presentations we generalize a result of Cautis-Kamnitzer-Morrison [arXiv:1210.6437v4]: Let $\mathfrak{g}$ be a reductive Lie algebra, and $A$ an…

This lecture reviews the classification of simple modules of double affine Hecke algebras via the K-theory of Steinberg varieties of affine type

We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally…

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…

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in…

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}^0_{\mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $\mathcal{D}$ in $\mathcal{C}^0_{\mathfrak{g}}$, we denote by…

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine…

B\"ohm and \c{S}tefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^\chi_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $\chi$…

We consider principal subspaces $W_{L(k\Lambda_0)}$ and $W_{N(k\Lambda_0)}$ of standard module $L(k\Lambda_0)$ and generalized Verma module $N(k\Lambda_0)$ at level $k\geq 1$ for affine Lie algebra of type $B_2^{(1)}$. By using the theory…

We study the category $\mathcal{C}$ generated by extremal weight modules over $U_q(\mathfrak{gl}_{>0})$. We show that $\mathcal{C}$ is a tensor category, and give an explicit description of the socle filtration of tensor product of any two…

We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…

We study the representation theory of the Lie superalgebra $\mathfrak{gl}(1|1)$, constructing two spectral sequences which eventually annihilate precisely the superdimension zero indecomposable modules in the finite-dimensional category.…

Recently, Meierfrankenfeld has published three theorems on the cohomology of a finitary module. They cover the local determination of complete reducibility; the local splitting of group extensions; and the representation of locally split…

The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a…

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is…

Let $\sg$ be a complex finite-dimensional simple Lie algebra and let $\sg_l$ be the corresponding generalized Takiff algebra. This paper studies the affine variety $\ssf+\sb_l$ where $\ssf$ is similar to a principal nilpotent element of…

In this paper we categorify the Heisenberg action on the Fock space via the category O of cyclotomic rational double affine Hecke algebras. This permits us to relate the filtration by the support on the Grothendieck group of O to a…

Given a weighted $\ell^2$ space with weights associated to an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a…