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Support varieties and representation type of small quantum groups
 Internat. Math. Res. Notices
"... Abstract. In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of su ..."
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Abstract. In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild. 1.
Constructing modules with prescribed cohomological support
, 2007
"... A cohomological support, Supp ∗ A (M), is defined for finitely generated modules M over a left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of Rmodules. It is proved that if the Amodule Ext ∗ R (M, M) is noetherian and Ext ∗ R (M, R) = 0 fo ..."
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Cited by 8 (0 self)
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A cohomological support, Supp ∗ A (M), is defined for finitely generated modules M over a left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of Rmodules. It is proved that if the Amodule Ext ∗ R (M, M) is noetherian and Ext ∗ R (M, R) = 0 for i ≫ 0, then every closed subset of Supp ∗ A (M) is the support of some finitely generated Rmodule. This theorem specializes to known realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. The theorem is also used to produce large families of finitely generated modules of finite projective dimension
Modules with prescribed cohomological support
, 2007
"... A cohomological support, Supp ∗ A (M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of Rmodules. It is proved that if the Amodule Ext ∗ R (M, M) is noetherian and Exti R (M, R) = 0 for ..."
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Cited by 7 (1 self)
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A cohomological support, Supp ∗ A (M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of Rmodules. It is proved that if the Amodule Ext ∗ R (M, M) is noetherian and Exti R (M, R) = 0 for i ≫ 0, then every closed subset of Supp ∗ A (M) is the support of some finitely generated Rmodule. This theorem is shown to specialize to realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. It is also used to produce large families of finitely generated modules of finite projective dimension
On injective modules and support varieties for the small quantum group
 International Mathematics Research Notices
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Tate and TateHochschild cohomology for finite dimensional Hopf Algebras
 J. Pure Appl. Algebra
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SUPPORT VARIETIES AND REPRESENTATION TYPE OF SELFINJECTIVE ALGEBRAS
"... Abstract. We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg [7]: If a finite dimensional selfinjective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on ..."
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Cited by 2 (1 self)
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Abstract. We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg [7]: If a finite dimensional selfinjective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in [23]. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small halfquantum groups, and Nichols (quantum symmetric) algebras.
TENSOR IDEALS AND VARIETIES FOR MODULES OF QUANTUM ELEMENTARY ABELIAN GROUPS
"... Abstract. In a previous paper we constructed rank and support variety theories for “quantum elementary abelian groups, ” that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a t ..."
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Abstract. In a previous paper we constructed rank and support variety theories for “quantum elementary abelian groups, ” that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties. 1.
RANK VARIETIES FOR HOPF ALGEBRAS
, 906
"... Abstract. We construct rank varieties for the Drinfel’d double of the Taft algebra and for uq(sl2). For the Drinfel’d double when n = 2 this uses a result which identifies a family of subalgebras that control projectivity of Λmodules whenever Λ is a Hopf algebra satisfying a certain homological con ..."
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Abstract. We construct rank varieties for the Drinfel’d double of the Taft algebra and for uq(sl2). For the Drinfel’d double when n = 2 this uses a result which identifies a family of subalgebras that control projectivity of Λmodules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext ∗ (M, M) is finitely generated over the cohomology ring of the Drinfel’d double for any finitelygenerated module M. 1.
Testing of Mathematical Group Structure Based on an Open Source Program (OSP)
, 2013
"... Abstract. Mathematical group structure is part of abstract algebra. It is useful in the development of solving method of problems that are abstract and difficult to test and represent through common algebraic operations. Thus, an open sourcebased computer program was developed to evaluate and prov ..."
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Abstract. Mathematical group structure is part of abstract algebra. It is useful in the development of solving method of problems that are abstract and difficult to test and represent through common algebraic operations. Thus, an open sourcebased computer program was developed to evaluate and prove a mathematical group structure. Several groups that are able to be tested and proved are aperiodic group, periodic group, mixed group, factor group, and normal subgroups. The developed program results showed that the evaluation of several types of the groups mentioned above performed well and perfect with a relatively shorter time than the manual test. In this program, the Java programming language was used since it can run well on several different operating system platforms. Besides, it is also able to be published free, so that programmers or other researchers can develop the application program further by adding more useful features.