Related papers: BGG category $\mathcal{O}$ and $\mathbb{Z}$-graded…

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…

The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting…

We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.

Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the…

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite…

We classify the blocks, compute the Verma flags of tilting and projective modules in the BGG category $\mathcal O$ for the exceptional Lie superalgebra $G(3)$. The projective injective modules in $\mathcal O$ are classified. We also compute…

Graded contractions of certain non-toral $\mathbb{Z}_2^3$-gradings on the simple Lie algebras $\mathfrak{so}(7,\mathbb C)$ and $\f{so}(8,\mathbb C)$ are classified up to two notions of equivalence. In particular, there arise two large…

A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, and a presentation of the category is given; specifically, we prove that seven relations among its four generating homomorphisms suffice to deduce all…

In the well-known settings of category theory enriched in a monoidal category V, the use of V-enriched functor categories and bifunctors demands that V be equipped with a symmetry, braiding, or duoidal structure. In this paper, we establish…

Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is…

In this paper, we discuss the generalization of finitary $2$-representation theory of finitary $2$-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple…

We construct a functor from the category of admissible finitely presented o-representations of GL(2,F) to the category of finite length o-representations of Gal_{Q_p}, for any finite extension F of Q_p and the ring of integers o of a finite…

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric…

We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of $\mathrm{SL}(2,\mathbb{R})$.

This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a…

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative…

We study the representation theory of a generalized graded Hecke algebra associated to a complex reflection group of type G(r,1,n), defined by Ram and Shepler. We use a realization of this algebra in the corresponding symplectic reflection…

We characterize finite-dimensional thick representations over ${\Bbb C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets.…

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical…

The goal of this paper is to relate the quantum category $\mathcal{O}$ (known also as the category of modules over the mixed quantum group) at an odd root of unity to the affine Hecke category. Namely, we prove equivalences of highest…