The Anick Complex and the Hochschild Cohomology of the Manturov (2,3)-Group

In this work, we find Hochschild cohomology groups of the Weyl associative conformal algebra with coefficients in all finite modules. The Weyl conformal algebra is the universal associative conformal envelope of the Virasoro Lie conformal algebra relative to the locality N = 2. In order to obtain this result, we adjust the algebraic discrete Morse ...

Added: June 6, 2023

Hochschild cohomology of Fermat type polynomials with non–abelian symmetries

Basalaev A., Ionov A., Journal of Geometry and Physics 2022 Vol. 174 Article 104450

For a polynomial f=x_1^n+…+x_N^n let Gf be the non–abelian maximal group of symmetries of f. This is a group generated by all g in GL(N,C), rescaling and permuting the variables, so that f(x)=f(g x). For any G subgroup in Gf we compute explicitly Hochschild cohomology of the category of G–equivariant matrix factorizations of f. We ...

Added: September 9, 2022

Cohomology Rings of the Plactic Monoid via a Grobner--Shirshov basis

Lopatkin V., Journal of Algebra and its Applications 2016 Vol. 15 No. 4 Article 1650082

In this paper, we calculate the cohomology ring and the Hochschild cohomology ring of the plactic monoid algebra via the Anick resolution using a Gröbner–Shirshov basis. ...

Added: October 29, 2021

Derivations of Leavitt path algebras

Lopatkin V., Journal of Algebra 2019 Vol. 520 P. 59–89

In this paper, we describe the K-module HH1(LK(Γ)) of outer derivations of the Leavitt path algebra LK(Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of HH1(LK(Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a ...

Added: October 29, 2021

Gröbner–Shirshov Bases, Normal Forms, Combinatorial and Decision Problems in Algebra

Bokut L., Chen Y., Kalorkoti K. et al., World Scientific, 2020.

The book is about (associative, Lie and other) algebras, groups, semigroups presented by generators and defining relations. They play a great role in modern mathematics. It is enough to mention the quantum groups and Hopf algebra theory, the Kac–Moody and Borcherds algebra theory, the braid groups and Hecke algebra theory, the Coxeter groups and semisimple ...

Added: September 27, 2021

The Gerstenhaber–Schack complex for prestacks

Van H. D., Lowen W., Advances in Mathematics 2018 Vol. 330 P. 173–228

The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex for an arbitrary prestack , we have to introduce a differential with an infinite sequence of components instead of ...

Added: September 13, 2018

Hochschild-Witt complex

Kaledin D., / Series arXiv "math". 2016.

In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex WCH∗(A) for any associative unital k-algebra A, with homology groups WHH∗(A). We prove that the group WHH0(A) ...

Added: May 18, 2016

Cohomology of exact categories and (non-)additive sheaves

Kaledin D., Lowen W., Advances in Mathematics 2015 Vol. 272 P. 652–698

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature. ...

Added: February 9, 2015

Height of exceptional collections and Hochschild cohomology of quasiphantom categories

Kuznetsov A., Journal fuer die reine und angewandte Mathematik 2015 Vol. 2015 No. 708 P. 213–243

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is ...

Added: December 22, 2013

Height of exceptional collections and Hochschild cohomology of quasiphantom categories

Kuznetsov A., / Series math "arxiv.org". 2012. No. 1211.4693.

Added: October 4, 2013

The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem

Markaryan N. S., Journal of London Mathematical Society 2009 Vol. 79 No. 79(2) P. 129–143

We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann–Roch theorem. ...

Added: October 12, 2012

Hochschild (co)homology of the second kind I

Polishchuk A., Positselski L., Transactions of the American Mathematical Society 2012 Vol. 364 No. 10 P. 5311–5368

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG category C of right CDG-modules over B, ...

Added: June 27, 2012