Varieties for Modules of Quantum Elementary Abelian Groups

By J. Pevtsova, S. Witherspoon. 2006

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We define a rank variety for a module of a noncocommutative Hopf algebra $A = \Lambda \rtimes G$ where $\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m$ and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba26896a522ff1e3fc5089e55e81c18fb5db2b5d

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In Algebra and Number Theory

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In Mathematical Physics

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Constructing modules with prescribed cohomological support (2007, background): “s an open problem whether similar ﬁniteness properties for cohomology hold when Ris not cocommutative; for positive solutions in interesting classes of such Hopf algebras see Pevtsova and Witherspoon [20], and the bibliography there. 4.3. Cohomological varieties. Let Rbe a Hopf algebra with rankkRﬁn”