Sarah Witherspoon
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Sarah Witherspoon has a total of 47 co-authors. Here are the top co-authors by number of shared publications:
- Anne V. Shepler
University of North Texas, Denton, USA
16 shared publications - Julia Pevtsova
Department of Mathematics, University of Washington, Seattle, USA
7 shared publications - Deepak Naidu
Department of Mathematics, Texas A&M University, College Station, Texas 77843
3 shared publications - Georgia Benkart
Department of Mathematics, University of Wisconsin-Madison, Madison, USA
3 shared publications - Van C. Nguyen
Department of Mathematics, Hood College, Frederick, MD, United States
3 shared publications - Aaron Lauve
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
2 shared publications
Latest Publications
- Open Access
Support varieties without the tensor product property
We show that over a perfect field, every non-semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties.
Open AccessSupport varieties without the tensor product property
We show that over a perfect field, every non-semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties.
- 1 March 2022
- Published by Springer Nature in Transformation Groups
- Vol. 27 (1) , 295-309
GERSTENHABER BRACKETS FOR SKEW GROUP ALGEBRAS IN POSITIVE CHARACTERISTIC
- 1 March 2022
- Published by Springer Nature in Transformation Groups
- Vol. 27 (1) , 295-309
GERSTENHABER BRACKETS FOR SKEW GROUP ALGEBRAS IN POSITIVE CHARACTERISTIC
- 20 January 2022
- Published by Springer Nature in Research in the Mathematical Sciences
- Vol. 9 (1) , 1-132
Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups
- 20 January 2022
- Published by Springer Nature in Research in the Mathematical Sciences
- Vol. 9 (1) , 1-132
Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups
- 16 January 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (2) , 1237-1257
Hochschild cohomology of twisted tensor products
- 16 January 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (2) , 1237-1257
Hochschild cohomology of twisted tensor products
- Open Access
New approaches to finite generation of cohomology rings
In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A'$ over a finite field, where $A$ can be viewed as a deformation of $A'$, and prove that if the finite generation condition holds for $A'$, then the same condition holds for $A$.Open AccessNew approaches to finite generation of cohomology rings
In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A'$ over a finite field, where $A$ can be viewed as a deformation of $A'$, and prove that if the finite generation condition holds for $A'$, then the same condition holds for $A$. - 11 August 2021
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 26 (1) , 97-116
Delta Sets and Polynomial Identities in Pointed Hopf Algebras
- 11 August 2021
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 26 (1) , 97-116
Delta Sets and Polynomial Identities in Pointed Hopf Algebras
- Open Access
Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups
We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.
Open AccessCohomology rings of finite-dimensional pointed Hopf algebras over abelian groups
We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.
- Open Access
Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products
We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.
Open AccessGerstenhaber brackets on Hochschild cohomology of general twisted tensor products
We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.
- Open Access
Support varieties for finite tensor categories: Complexity, realization, and connectedness
We advance support variety theory for finite tensor categories. First we show that the dimension of the support variety of an object equals the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension. Then we show that every conical subvariety of the support variety of the unit object may be realized as the support variety of an object. Finally, we show that the support variety of an indecomposable object is connected.
Open AccessSupport varieties for finite tensor categories: Complexity, realization, and connectedness
We advance support variety theory for finite tensor categories. First we show that the dimension of the support variety of an object equals the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension. Then we show that every conical subvariety of the support variety of the unit object may be realized as the support variety of an object. Finally, we show that the support variety of an indecomposable object is connected.
- Open Access
Hochschild cohomology of twisted tensor products
For a tensor product of algebras twisted by a bicharacter, we completely describe its Hochschild cohomology, as a Gerstenhaber algebra, in terms of the Hochschild cohomology of its component parts. This description generalizes a result of Bergh and Oppermann. It allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology for a number of new examples.
Open AccessHochschild cohomology of twisted tensor products
For a tensor product of algebras twisted by a bicharacter, we completely describe its Hochschild cohomology, as a Gerstenhaber algebra, in terms of the Hochschild cohomology of its component parts. This description generalizes a result of Bergh and Oppermann. It allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology for a number of new examples.