Hochschild cohomology of Fermat type polynomials with non–abelian symmetries

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

Download Full-text

Dimer Models and Hochschild Cohomology

Journal of Algebra ◽

10.1016/j.jalgebra.2021.06.003 ◽

2021 ◽

Author(s):

Michael Wong

Keyword(s):

Hochschild Cohomology ◽

Dimer Models

Download Full-text

Twisted Homology of Quantum SL(2) - Part II

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is009009022jkt091 ◽

2009 ◽

Vol 6 (1) ◽

pp. 69-98 ◽

Cited By ~ 5

Author(s):

Tom Hadfield ◽

Ulrich Krähmer

Keyword(s):

Noncommutative Geometry ◽

Hochschild Cohomology ◽

Cyclic Homology ◽

Coordinate Ring ◽

Volume Form ◽

Twisted Homology

AbstractWe complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

Download Full-text

A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15240 ◽

2013 ◽

Vol 112 (2) ◽

pp. 182 ◽

Cited By ~ 1

Author(s):

Shoham Shamir

Keyword(s):

Spectral Sequence ◽

Loop Space ◽

Hochschild Cohomology ◽

Derived Category ◽

Orientable Manifold ◽

Support Varieties ◽

Mod P

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

Download Full-text

Comparison morphisms between two projective resolutions of monomial algebras

Revista de la Unión Matemática Argentina ◽

10.33044/revuma/v59n1a01 ◽

2017 ◽

pp. 1-31

Author(s):

María Julia Redondo ◽

Lucrecia Román

Keyword(s):

Lie Algebra ◽

Hochschild Cohomology ◽

Cohomology Groups ◽

Efficient Tool ◽

Simple Description ◽

Monomial Algebra ◽

Cup Product ◽

Lie Bracket ◽

Projective Resolutions ◽

Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Download Full-text