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DIMENSIONS OF TRIANGULATED CATEGORIES VIA KOSZUL OBJECTS
"... Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin alge ..."
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Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras. 1.
COHOMOLOGY OF FINITE DIMENSIONAL POINTED HOPF ALGEBRAS
"... Abstract. We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Ex ..."
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Abstract. We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig’s small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite dimensional pointed Hopf algebra and its related Nichols algebra. 1.
COHOMOLOGY AND SUPPORT VARIETIES FOR LIE SUPERALGEBRAS
, 2006
"... 1.1. The blocks of the Category O (or relative Category OS) for complex semisimple Lie algebras are well known examples of highest weight categories, as defined in [CPS], with finitely many simple modules. These facts imply that the projective resolutions for modules in these categories have finite ..."
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1.1. The blocks of the Category O (or relative Category OS) for complex semisimple Lie algebras are well known examples of highest weight categories, as defined in [CPS], with finitely many simple modules. These facts imply that the projective resolutions for modules in these categories have finite length, so the cohomology (or extensions) can be nonzero in
VARIETIES FOR MODULES OF QUANTUM ELEMENTARY ABELIAN GROUPS
"... Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ ⋊ G where Λ = k[X1,..., Xm]/(X ℓ 1,..., X ℓ m), G = (Z/ℓZ) 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 ..."
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Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ ⋊ G where Λ = k[X1,..., Xm]/(X ℓ 1,..., X ℓ m), G = (Z/ℓZ) 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 Λmodules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λmodules coincide with those of Erdmann and Holloway. 1.
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.
The restricted nullcone
 Cont. Math
"... Abstract. Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p> 0. The restricted nullcone of g, denoted by N1(g), consists of those x ∈ g such that x [p] = 0. In this paper the authors provide a concrete description of this variety (via closures of Ric ..."
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Abstract. Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p> 0. The restricted nullcone of g, denoted by N1(g), consists of those x ∈ g such that x [p] = 0. In this paper the authors provide a concrete description of this variety (via closures of Richardson orbits) when g is the Lie algebra of a reductive group G and p a good prime. Various applications to representation theory and cohomology theory are provided. 1.
BIFUNCTOR COHOMOLOGY AND COHOMOLOGICAL FINITE GENERATION FOR REDUCTIVE GROUPS
, 2010
"... Let G be a reductive linear algebraic group over a field k.LetA be a finitely generated commutative kalgebra on which G acts rationally by kalgebra automorphisms. Invariant theory states that the ring of invariants A G = H 0 (G, A) is finitely generated. We show that in fact the full cohomology ri ..."
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Let G be a reductive linear algebraic group over a field k.LetA be a finitely generated commutative kalgebra on which G acts rationally by kalgebra automorphisms. Invariant theory states that the ring of invariants A G = H 0 (G, A) is finitely generated. We show that in fact the full cohomology ring H ∗ (G, A) is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed in [22]. We also continue the study of bifunctor cohomology of Ɣ ∗ (gl (1)).
Generic and Maximal Jordan types
"... Abstract. For a finite group scheme G over a field k of characteristic p> 0, we associate new invariants to a finite dimensional kGmodule M. Namely, for each generic point of the projectivized cohomological variety Proj H • (G, k) we exhibit a “generic Jordan type ” of M. In the very special cas ..."
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Abstract. For a finite group scheme G over a field k of characteristic p> 0, we associate new invariants to a finite dimensional kGmodule M. Namely, for each generic point of the projectivized cohomological variety Proj H • (G, k) we exhibit a “generic Jordan type ” of M. In the very special case in which G = E is an elementary abelian pgroup, our construction specializes to the nontrivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the nonmaximal support variety Γ(G)M, a closed subset of Proj H • (G, k) which is proper even when the dimension of M is not divisible by p. Elementary abelian psubgroups of a finite group G capture significant aspects of the cohomology and representation theory of G. For example, if k is a field of characteristic p> 0, then a theorem of D. Quillen [18] asserts that the Krull dimension of the cohomology algebra H • (G, k) is equal to the maximum of the ranks of
Constructions for infinitesimal group schemes
, 2010
"... Let G be an infinitesimal group scheme over a field k of characteristic p> 0. We introduce the global pnilpotent operator ΘG: k[G] → k[V (G)], where V (G) is the scheme which represents 1parameter subgroups of G. This operator ΘG applied to M encodes the local Jordan type of M, and leads to c ..."
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Let G be an infinitesimal group scheme over a field k of characteristic p> 0. We introduce the global pnilpotent operator ΘG: k[G] → k[V (G)], where V (G) is the scheme which represents 1parameter subgroups of G. This operator ΘG applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain kGmodules M (including those of constant Jordan type), we employ ΘG to associate various algebraic vector bundles on P(G), the projectivization of V (G). These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on P(G).
Representation Theory Of Noetherian Hopf Algebras Satisfying A Polynomial Identity
, 1997
"... . A class of Noetherian Hopf algebras satisfying a polynomial identity is axiomatised and studied. This class includes group algebras of abelianbyfinite groups, finite dimensional restricted Lie algebras, and quantised enveloping algebras and quantised function algebras at roots of unity. Some ..."
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. A class of Noetherian Hopf algebras satisfying a polynomial identity is axiomatised and studied. This class includes group algebras of abelianbyfinite groups, finite dimensional restricted Lie algebras, and quantised enveloping algebras and quantised function algebras at roots of unity. Some common homological and representationtheoretic features of these algebras are described, with some indications of recent and current developments in research on each of the exemplar classes. It is shown that the finite dimensional representation theory of each of these algebras H reduces to the study of a collection Alg(H) of (finite dimensional) Frobenius algebras. The properties of this family of finite dimensional algebras are shown to be intimately connected with geometrical features of central subHopf algebras of H. A number of open questions are listed throughout. 1. Introduction My aim in this paper is to review some common properties exhibited by four large and important cla...