Related papers: Representation theory of Geigle-Lenzing complete i…

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an…

This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster)…

We discuss finitely graded Iwanaga-Gorenstein (IG) algebras $A$ and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where $A$ is…

We study quantized enveloping algebras called twisted Yangians. They are analogues of the Yangian Y(gl(N)) for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in Y(gl(N)) and coideals with respect to…

In this paper we study general quantum affinizations $\U_q(\hat{\Glie})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1)…

Let $\Lambda$ be an Artin algebra and ${\mathsf{mod}}\mbox{-} ({\underline{\mathsf{Gprj}}}\mbox{-}\Lambda)$ the category of finitely presented functors over the stable category ${\underline{\mathsf{Gprj}}}\mbox{-}\Lambda$ of finitely…

Let $(R, \m, k)$ be a complete Cohen-Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen-Macaulay module, called $\uh$-length. Among other results, it is proved that, for a given balanced big…

This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of…

Quivers (directed graphs) and species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their…

Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of a…

To each irreducible infinite dimensional representation $(\pi,\cH)$ of a $C^*$-algebra $\cA$, we associate a collection of irreducible norm-continuous unitary representations $\pi_{\lambda}^\cA$ of its unitary group $\U(\cA)$, whose…

We provide a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson-Ginzburg-Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories…

An Artin algebra $\Lambda$ is said to be of finite Cohen-Macaulay type, $\rm{CM}$-finite for short, if the full subcategory $\rm{Gprj}\mbox{-} \Lambda$ of finitely generated Gorenstein projective $\Lambda$-modules is of finite…

In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions.…

We present a simple description of moduli spaces of torsion-free D-modules (``D-bundles'') on general smooth complex curves X, generalizing the identification of the space of ideals in the Weyl algebra with Calogero-Moser quiver varieties.…

Let $\Lambda$ be a finite-dimensional algebra with finite global dimension, $R_k=K[X]/(X^k)$ be the $\mathcal{Z}$-graded local ring with $k\geq1$, and $\Lambda_k=\Lambda\otimes_K R_k$. We consider the singularity category…

We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group…

We describe the dimensions of Hochschild (co)homology groups of weighted projective curves over complex numbers. Surprisingly, all but one of those numbers depend only on the genus of the underlying non-weighted curve and the number of…

We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space…

In a previous work the authors gave a conceptual explanation for the linearity of the Weil representation over a finite field k of odd characteristic: There exists a canonical system of intertwining operators between the Lagrangian models…