scholarly journals Finite groups acting linearly: Hochschild cohomology and the cup product

2011 ◽  
Vol 226 (4) ◽  
pp. 2884-2910 ◽  
Author(s):  
Anne V. Shepler ◽  
Sarah Witherspoon
Keyword(s):  
Finite Groups ◽  
Hochschild Cohomology ◽  
Cup Product

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

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 $\operatorname{HH}^{\ast} (A)$.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

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 $\operatorname{HH}^{\ast} (A)$.

scholarly journals Hochschild cohomology and Linckelmann cohomology for blocks of finite groups

2003 ◽  
Vol 178 (1) ◽  
pp. 87-100 ◽  
Author(s):  
Jonathan Pakianathan ◽  
Sarah Witherspoon
Keyword(s):  
Finite Groups ◽  
Hochschild Cohomology

scholarly journals Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Constantin-Cosmin Todea
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Finite Groups ◽  
Hochschild Cohomology ◽  
Product Formula ◽  
Group Algebras ◽  
Closed Field ◽  
Additive Decomposition ◽  
A Finite Group ◽  
Block Algebra

AbstractWe give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a product formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For an algebraically closed field

scholarly journals On the geometry of lattices and finiteness of Picard groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Florian Eisele
Keyword(s):  
Finite Groups ◽  
Hochschild Cohomology ◽  
Outer Automorphism ◽  
Modular System ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Picard Groups ◽  
Finite Dimensional ◽  
Varieties Of Lattices ◽  
Definition Of

Abstract Let ( K , 𝒪 , k ) {(K,\mathcal{O},k)} be a p-modular system with k algebraically closed and 𝒪 {\mathcal{O}} unramified, and let Λ be an 𝒪 {\mathcal{O}} -order in a separable K-algebra. We call a Λ-lattice L rigid if Ext Λ 1 ⁡ ( L , L ) = 0 {{\operatorname{Ext}}^{1}_{\Lambda}(L,L)=0} , in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into “varieties of lattices”, we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over 𝒪 {\mathcal{O}} are always finite.

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