[Paper Review] Hopf algebra structures and tensor products for group algebras

Jon Carlson, Srikanth B. Iyengar|arXiv (Cornell University)|Dec 14, 2016

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper establishes that for the group algebra of an elementary abelian $p$-group, the action of the polynomial subring $S$ of the cohomology ring—generated by Bocksteins of degree-one elements—on module cohomology is independent of the choice between the group-like and Lie-like Hopf algebra coalgebra structures. This invariance ensures that tensor products involving modules $L_\zeta$ for $\zeta \in S$ are well-defined regardless of the coalgebra structure, resolving a key obstruction in relating group representations to commutative algebra via Hopf algebra structures.

ABSTRACT

The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of commutative restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine cup products on cohomology of modules. However, it is proved in this paper that the products with elements of the polynomial subring of the cohomology ring generated by the Bocksteins of the degree one elements are independent of the choice of these coalgebra structures.

Motivation & Objective

• To resolve the ambiguity in tensor product constructions for $kE$-modules arising from different Hopf algebra coalgebra structures on the group algebra of an elementary abelian $p$-group.
• To show that the action of the polynomial subring $S$—generated by Bocksteins of degree-one elements—on $\operatorname{Ext}^*_{kE}(M,M)$ is independent of the coalgebra structure.
• To establish that the isomorphism class of $L_\zeta \otimes_k M$ is well-defined for $\zeta \in S$, regardless of whether the group or Lie coalgebra structure is used.
• To support categorical equivalences between commutative algebra and group representation theory by showing that the critical map $\theta^M$ factors through Hochschild cohomology, with the $S$-restriction being independent of coalgebra choice.

Proposed method

• Utilizes the factorization of the cohomology action map $\theta^M: \operatorname{Ext}^*_{kE}(k,k) \to \operatorname{Ext}^*_{kE}(M,M)$ through Hochschild cohomology $\operatorname{HH}^*(kE/k; kE)$, separating coalgebra dependence from module dependence.
• Employs Sweedler’s notation for comultiplication and defines the diagonal action on tensor products of modules via $\alpha \cdot (x \otimes m) = \sum_{(\alpha)} \alpha_1 x \otimes \alpha_2 m$.
• Applies the isomorphism $kE \cong k[x_1,\dots,x_r]/(x_1^p,\dots,x_r^p)$ to realize the group algebra as a restricted enveloping algebra of a commutative restricted Lie algebra with trivial bracket and $p$-power operation.
• Uses the fact that $E$ is a direct product of cyclic $p$-groups to perform explicit calculations showing that the Hochschild cohomology map is identical on $S$ for both coalgebra structures.
• Relies on the result of Pevtsova and Witherspoon that $\theta^M$ factors through $\operatorname{HH}^*(kE/k; kE)$, reducing the problem to comparing the first map in the factorization.
• Analyzes the behavior of $L_\zeta$-modules under tensor products using both coalgebra structures, with explicit basis computations in characteristic 2 to demonstrate non-invariance outside $S$.

Experimental results

Research questions

• RQ1 Does the action of the cohomology ring on $\operatorname{Ext}^*_{kE}(M,M)$ depend on the choice of Hopf algebra coalgebra structure for the group algebra of an elementary abelian $p$-group?
• RQ2 Is the action of the polynomial subring $S$, generated by Bocksteins of degree-one elements, invariant under the choice of coalgebra structure?
• RQ3 Can the isomorphism class of $L_\zeta \otimes_k M$ be defined independently of the coalgebra structure when $\zeta \in S$?
• RQ4 Does the support variety of a module remain unchanged when the coalgebra structure is altered, provided the restriction to $S$ is preserved?
• RQ5 Is there a strong converse to the invariance result, such that tensor products are isomorphic only when the cohomology class lies in $S$?

Key findings

• The action of the subring $S \subseteq \operatorname{H}^*(E,k)$—generated by Bocksteins of degree-one elements—on $\operatorname{Ext}^*_{kE}(M,M)$ is independent of the choice between the group-like and Lie-like Hopf algebra coalgebra structures on $kE$.
• As a consequence, the isomorphism class of the tensor product $L_\zeta \otimes_k M$ is well-defined for any $kE$-module $M$ and any $\zeta \in S$, regardless of the coalgebra structure used.
• The Hochschild cohomology map $\operatorname{HH}^*(kE/k; kE) \to \operatorname{Ext}^*_{kE}(M,M)$, which factors through $\theta^M$, has the same restriction to $S$ under both coalgebra structures, ensuring invariance.
• The support variety $V_G(M)$ is independent of the coalgebra structure because $J(M) \cap S$—the annihilator of $\operatorname{Ext}^*_{kE}(M,M)$ in $S$—is invariant under the choice of coalgebra.
• In characteristic 2, explicit computations show that $L_\zeta \otimes_k L_\zeta$ is decomposable under the Lie coalgebra structure but indecomposable under the group coalgebra structure when $\zeta \notin S$, demonstrating the sharpness of the result.
• Computer experiments using Magma suggest that the isomorphism class of $L_{\gamma_1} \otimes_k L_{\gamma_2}$ is independent of coalgebra structure if and only if at least one of $\gamma_1, \gamma_2$ lies in $S$, supporting a strong converse to the main theorem.