Deforming group actions on Koszul algebras

Shepler, Anne V.; Witherspoon, Sarah

doi:10.4310/mrl.2018.v25.n5.a12

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…In this section, we show that the group-twisted Alexander-Whitney and Eilenberg-Zilber maps provide a way to convert between the reduced bar resolution for a skew group algebra S ⋊ G and a choice of twisted product resolution for S ⋊ G, which generally can be much smaller. Compare to [22,Lemma 4.4] for Koszul algebras; our result here is stated for more general algebras S, and we provide a proof based on the explicit chain maps AW G and EZ G defined above. We will see in the next section that the maps AW G and EZ G have deformation-theoretic consequences.…”

Section: Conversion Between Resolutionsmentioning

confidence: 90%

“…We explain here how Theorem 4.1 gives a unified proof of this connection using the group-twisted Alexander-Whitney and Eilenberg-Zilber maps. This approach is more elegant than the somewhat ad hoc proofs given in [21,22]. It also more easily applies to other settings.…”

Section: Applications To Deformation Theorymentioning

confidence: 97%

“…We now consider some applications of Theorem 4.1. We used Hochschild cohomology to analyze PBW deformations of S ⋊ G in [21] and [22], particularly in the case that S = S(V ) is a symmetric algebra. Our Theorem 4.1 generalizes both [21, Lemma 7.3] and [22,Lemma 4.4], which provide an explicit connection between Hochschild cocycles and PBW deformations.…”

Section: Applications To Deformation Theorymentioning

confidence: 99%

“…We used Hochschild cohomology to analyze PBW deformations of S ⋊ G in [21] and [22], particularly in the case that S = S(V ) is a symmetric algebra. Our Theorem 4.1 generalizes both [21, Lemma 7.3] and [22,Lemma 4.4], which provide an explicit connection between Hochschild cocycles and PBW deformations. We explain here how Theorem 4.1 gives a unified proof of this connection using the group-twisted Alexander-Whitney and Eilenberg-Zilber maps.…”

Section: Applications To Deformation Theorymentioning

confidence: 99%

“…Similarly, in [22], we consider the case when S is a quadratic algebra, and especially the case when S is Koszul. Again, we used an explicit chain map established with improvised methods (see [22,Lemmas 4.4 and 4.6]) to convert between resolutions and study explicit conditions giving PBW deformations of S ⋊ G. Indeed, the proof of [22,Theorem 2.5] relies on explicit information provided by chain maps. Here again, chain maps may be taken from Theorem 4.1 instead to give a more conceptual view of the deformation-theoretic results in [22].…”

Section: Applications To Deformation Theorymentioning

confidence: 99%

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Group-twisted Alexander-Whitney and Eilenberg-Zilber maps

Shepler¹,

Witherspoon²

2019

Preprint

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We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted chain maps allow us to transfer information between resolutions for use in homology theories, for example, in those governing deformation theory. We show how to translate in particular from the default (but often cumbersome) reduced bar resolution to a more convenient twisted product resolution. This provides a more universal approach to some known results classifying PBW deformations.

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Section: Conversion Between Resolutionsmentioning

confidence: 90%

Section: Applications To Deformation Theorymentioning

confidence: 97%

Section: Applications To Deformation Theorymentioning

confidence: 99%

Section: Applications To Deformation Theorymentioning

confidence: 99%

Section: Applications To Deformation Theorymentioning

confidence: 99%

See 3 more Smart Citations

Group-twisted Alexander-Whitney and Eilenberg-Zilber maps

Shepler¹,

Witherspoon²

2019

Preprint

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Group-Twisted Alexander-Whitney and Eilenberg-Zilber Maps

Shepler

Witherspoon

2020

Algebr Represent Theor

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Alexander-Whitney and Eilenberg-Zilber maps traditionally convert between the tensor product of standard resolutions and the standard resolution of a tensor product of algebras. We examine Alexander-Whitney and Eilenberg-Zilber maps for twisted tensor products, which include skew group algebras, smash products of Hopf algebras, Ore extensions, and universal enveloping algebras. These maps convert between the twist of standard resolutions and the standard resolution of a twist. We extend these to chain maps to and from twists of other resolutions. This allows one to transfer homological information between various resolutions of algebras and to expedite results on the deformation theory of twisted tensor product algebras.

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