Related papers: Verma Howe duality and LKB representations

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an…

We study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel'd doubles of rank one pointed Hopf algebras of nilpotent type. We study, in particular, under what conditions a…

We first formulate a definition of tensor product for two modules for a vertex operator algebra in terms of a certain universal property and then we give a construction of tensor products. We prove the unital property of the adjoint module…

We establish a duality between monads and monadic morphisms in any $(\infty,2)$-category and characterize monadic morphisms in a wide class of examples. This duality unifies several dualities between algebraic structures and their…

Let $(R^{\vee},R)$ be a dual pair of Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra $H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter-Drinfeld…

We establish a duality relation between one of the twisted group algebras of the hyperoctahedral groupf H_k and a Lie superalgebra q(n_0) \oplus q(n_1) for any integers k and n_0, n_1, where q(n_0) and q(n_1) denote the ``queer''…

Using the fusion product of the representations of the Lie algebra $\mathfrak{sl}_2$ we construct a set of the integrable highest weight $\hat{\mathfrak{sl}_2}$-modules $L^D$, depending on the vector $D\in\mathbb{N}^{k+1}$. In a special…

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well…

Based on previous results on the classification of finite-dimensional Nichols algebras over dihedral groups and the characterization of simple modules of Drinfeld doubles, we compute the irreducible characters of the Drinfeld doubles of…

Given a suitable ordering of the positive root system associated with a semisimple Lie algebra, there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module…

We study the highest weight and continuous tensor product representations of q-deformed Lie algebras through the mappings of a manifold into a locally compact group. As an example the highest weight representation of the q-deformed algebra…

The category of weight modules $L_k(\mathfrak{sl}_2)\text{-wtmod}$ of the simple affine vertex algebra of $\mathfrak{sl}_2$ at an admissible level $k$ is neither finite nor semisimple and modules are usually not lower-bounded and have…

The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of GL(n). Using the hive model for the…

We present a quick approach to computing the $K$-theory of the category of locally compact modules over any order in a semisimple $\mathbb{Q}$-algebra. We obtain the $K$-theory by first quotienting out the compact modules and subsequently…

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…

For a semisimple Lie algebra $\mathfrak{g}$ of rank $n$, let $\overline{U}_\zeta(\mathfrak{g})$ be the restricted quantum group of $\mathfrak{g}$ at a primitive fourth root of unity. This quantum group admits a natural Borel-induced…

In this paper we study the relative tensor product of module categories over braided fusion categories using, in part, the notion of the relative center of a module category. In particular we investigate the canonical tensor category…

Field theories with kinematic Lie algebras, such as field theories featuring colour-kinematics duality, possess an underlying algebraic structure known as BV${}^{\color{gray} \blacksquare}$-algebra. If, additionally, matter fields are…

We study the category $\cal I_{\gr}$ of graded representations with finite--dimensional graded pieces for the current algebra $\lie g\otimes\bc[t]$ where $\lie g$ is a simple Lie algebra. This category has many similarities with the…

Let $G$ be a simple algebraic group in defining characteristic $p>0$, and let $V$ be an irreducible $G$-module which is the tensor product of exactly two non-trivial modules. We obtain a criterion for $V$ to have the zero weight. In…