Group actions on algebras and the graded Lie structure of Hochschild cohomology

Abstract. We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeldtype deformation. We fit all these deformations into a general theory, connecting Poincaré-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

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“…Next we generalize [23,Theorem 8.7] to fields of arbitrary characteristic (see also [25,Theorem 11.4…”

Section: Since [λ λ] = 2dmentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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“…Representations and homological properties of skew group algebras (or more generally, group-graded algebras and crossed products) have been widely studied; see [5,7,16,[19][20][21]27]. When |G|, the order of G, is invertible in k, it has been shown that ΛG and Λ share many common properties.…”

Section: Introductionmentioning

confidence: 99%

Representations of modular skew group algebras

2015

Trans. Amer. Math. Soc.

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In this paper we study representations of skew group algebras ΛG, where Λ is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field k with characteristic p 0, and G is an arbitrary finite group each element of which acts as an algebra automorphism on Λ. We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for p = 2, 3. When Λ is a locally finite graded algebra and the action of G on Λ preserves grading, we show that ΛG is a generalized Koszul algebra if and only if Λ is.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 6294 LIPING LI category, etc., and Λ is the corresponding path algebra, incidence algebra, or category algebra. Under this assumption, we show that if ΛS is of finite representation type (for the situation where p 5 and Λ is not a local algebra) or of finite global dimension, then the action of S on E must be free. In this situation, ΛS is the matrix algebra over Λ S , the subalgebra of Λ constituted of all elements in Λ fixed by the action of S. Therefore, ΛS is Morita equivalent to Λ S , and we can prove:closed under the action of S and n i=1 e i = 1. Then: (1) gldim ΛG < ∞ if and only if gldim Λ < ∞ and S acts freely on E. Moreover, if gldim ΛG < ∞, then gldim ΛG = gldim Λ. (2) ΛG is an Auslander algebra if and only if Λ is and S acts freely on E. (3) Suppose that p = 2, 3 and Λ is not a local algebra. Then ΛG is of finite representation type if and only if so is Λ S and S acts freely on E. If Λ ∼ = Λ S ⊕ B as Λ S -bimodules, ΛG has finite representation type if and only if so does Λ and S acts on E freely.Let P be a finite connected poset on which every element in G acts as an automorphism. The Grothendick construction T = G ∝ P is called a transporter category. It is a finite EI category, i.e., every endomorphism in T is an isomorphism. Representations of transporter categories and finite EI categories have been studied in [10,11,22,23,25,26]. In a paper [26], Xu showed that the category algebra kT is a skew group algebra of the incidence algebra kP. We study its representation type for p = 2, 3, and it turns out that we can only get finite representation type for very few cases: Theorem 1.2. Let G be a finite group acting on a connected finite poset P and suppose that p = 2, 3. Then the transporter category T = G ∝ P is of finite representation type if and only if one of the following conditions is true:(1) |G| is invertible in k and P is of finite representation type;(2) P has only one element and G is of finite representation type.When Λ is a locally finite graded algebra and the action of G preserves grading, the skew group algebra ΛG has a natural grading. Therefore, it is reasonable to characterize its Koszul property. Since |G| might not be invertible in k, the degree 0 part of ΛG might not be semisimple, and the classical Koszul theory cannot apply. Thus we use ...

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“…In the lemma below, we give a formula for the

ϕ

‐circle product of special types of elements. Our formula may be compared with [, Lemma 4.1; , Theorem 7.2]. Due to the structure of the Hochschild cohomology of

S (V) # G

stated in Proposition , this formula will in fact suffice to compute all brackets.…”

Section: ϕ‐Circle Product Formula and Projections Onto Group Componentsmentioning

confidence: 99%

“…In Corollary we also show that the Hochschild cohomology

{prefixHH}^{•} false(S (V) # G false)

is a graded Gerstenhaber algebra with respect to a certain natural grading coming from the geometry of the fixed spaces

V^{g}

, which we refer to as the codimension grading. Section 6 consists of examples and a general explanation of (non)vanishing of the Gerstenhaber bracket for

S (V) # G

, rephrasing some of the results of .…”

Section: Introductionmentioning

confidence: 99%

The Gerstenhaber bracket as a Schouten bracket for polynomial rings extended by finite groups

Negron

Witherspoon

2017

Proc. London Math. Soc.

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Abstract. We apply new techniques to compute Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets on polyvector fields. We obtain as consequences some conditions under which brackets are always 0, and show that the Hochschild cohomology is a graded Gerstenhaber algebra under the codimension grading, strengthening known results.

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Group actions on algebras and the graded Lie structure of Hochschild cohomology

Cited by 26 publications

References 23 publications

PBW Deformations of Skew Group Algebras in Positive Characteristic

PBW Deformations of Skew Group Algebras in Positive Characteristic

Representations of modular skew group algebras

The Gerstenhaber bracket as a Schouten bracket for polynomial rings extended by finite groups

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