Related papers: $A_{\infty}$-algebra Structures Associated to $\ma…

We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version…

We study a categorified generalization of Koszul duality that treats duality phenomena among monoidal categories. We establish Koszul duality results for stable monoidal infinity-categories associated with Artin algebras and related…

Bialgebras associated to Yang-Baxter operators satisfying the Hecke equation, are considered. It is shown that they are Koszul algebras. Their Poincare' series are calculated via the Poincare' series of the corresponding quantum spaces.

We introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application, we prove that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. We…

We construct a generalization of Koszul duality in the sense of Keller--Lef\`evre for not necessarily augmented algebras. This duality is closely related to classical Morita duality and specializes to it in certain cases.

Let $M$ be a compact manifold. and $D$ a Dirac type differential operator on $M$. Let $A$ be a $C^*$-algebra. Given a bundle $W$ of $A$-modules over $M$ (with connection), the operator $D$ can be twisted with this bundle. One can then use a…

Let S=K[x_1,...,x_n] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f_1,f_2,...,f_k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and…

We study algebraic structures ($L_\infty$ and $A_\infty$-algebras) introduced by Gaiotto, Moore and Witten in their recent work devoted to certain supersymmetric 2-dimensional massive field theories. We show that such structures can be…

We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key…

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on…

In the preprint of V. Bardakov, T. Kozlovskaya, D. Talalaev (Self-distributive bialgebras, arXiv:2501.19152) it was formulated a problem of classification of self-distributive bialgebras and was given classification of two-dimensional…

We motivate and study the reduced Koszul map, relating the invariant bilinear maps on a Lie algebra and the third homology. We show that it is concentrated in degree 0 for any grading in a torsion-free abelian group, and in particular it…

This paper has two parts. The main goal, carried out in Part I, is to survey some recent work by the authors in which "forced" grading constructions have played a significant role in the representation theory of semisimple algebraic groups…

We give a description of the connected graded algebras which are finitely generated and presented of global dimension 2 or 3 and which are Gorenstein. These algebras are constructed from multilinear forms. We generalize the construction by…

It is shown that the closure of the infinitesimal symmetry transformations underlying classical ${\cal W}$ algebras give rise to L$_\infty$ algebras with in general field dependent gauge parameters. Therefore, the class of well understood…

In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference,…

For any Legendrian knot or link in $\mathbb{R}^3$, we construct an $L_\infty$ algebra that can be viewed as an extension of the Chekanov-Eliashberg differential graded algebra. The $L_\infty$ structure incorporates information from rational…

Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth…

We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an $\infty$-category structure to the category of curved Lie algebras over an algebraically…

The notion of PROP models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. We prove a Koszul duality theory for PROPs generalizing the one for…