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16
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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LieRinehart algebras, Gerstenhaber algebras, and
, 1997
"... For any LieRinehart algebra (A, L), (i) generators for the Gerstenhaber algebra ΛAL correspond bijectively to right (A, L) connections on A in such a way that (ii) B(atalin)V(ilkovisky) structures correspond bijectively to right (A, L)module structures on A. When L is projective as an Amodule, ..."
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Cited by 29 (10 self)
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For any LieRinehart algebra (A, L), (i) generators for the Gerstenhaber algebra ΛAL correspond bijectively to right (A, L) connections on A in such a way that (ii) B(atalin)V(ilkovisky) structures correspond bijectively to right (A, L)module structures on A. When L is projective as an Amodule, given an exact generator ∂, the homology of the BV algebra (ΛAL, ∂) coincides with the homology of L with coefficients in A with reference to the right (A, L)module structure determined by ∂. When L is also of finite rank n, there are bijective correspondences between (A, L)connections on Λn AL and right (A, L)connections on A and between left (A, L)module structures on Λn AL and right (A, L)module structures on A. Hence L and generators there are bijective correspondences between (A, L)connections on Λn A for the Gerstenhaber bracket on ΛAL and between (A, L)module structures on Λn AL and BV algebra structures on ΛAL. The homology of such a BV algebra (ΛAL, ∂) coincides with the cohomology of L with coefficients in Λn AL, with reference to the left (A, L)module structure determined by ∂. Some applications to Poisson structures and to differential geometry are discussed.
CURRENT ALGEBRAS, HIGHEST WEIGHT CATEGORIES AND QUIVERS
, 2007
"... Abstract. We study the category of graded finitedimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions ..."
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Cited by 7 (3 self)
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Abstract. We study the category of graded finitedimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions betweeen simple objects. The simple objects in the category are parametized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on affine the weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and tame type that arise from our study. We also prove that the path algebra of star shaped quivers are the Extalgebra of a suitable subcategory.
On Support Varieties Of AuslanderReiten Components
 Indag. Math
"... Let u(L; Ø) be the reduced enveloping algebra associated to a finite dimensional restricted Lie algebra (L; [p]) and a linear form Ø 2 L . It is shown that a connected component \Theta of the stable AuslanderReiten quiver of u(L; Ø) is of type Z[A1 ], whenever its support variety VL (\Theta) has ..."
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Cited by 4 (4 self)
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Let u(L; Ø) be the reduced enveloping algebra associated to a finite dimensional restricted Lie algebra (L; [p]) and a linear form Ø 2 L . It is shown that a connected component \Theta of the stable AuslanderReiten quiver of u(L; Ø) is of type Z[A1 ], whenever its support variety VL (\Theta) has dimension 3. Various applications concerning ARcomponents of Lie algebras of algebraic groups and the structure of hearts of principal indecomposable u(L; Ø)modules are given. 1. Introduction and Preliminaries In recent work [10] K. Erdmann has shown that the nonperiodic components of the stable AuslanderReiten quivers belonging to wild blocks of a modular group algebra are isomorphic to Z[A1 ]. Accordingly, the AuslanderReiten theory of group algebras is now very well understood. By contrast, relatively little is known about the ARquivers of the family (u(L; Ø)) Ø2L of reduced enveloping algebras associated to a restricted Lie algebra (L; [p]). One main problem in this context is...
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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Cited by 4 (1 self)
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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
Irreducible characters of general linear superalgebra and super duality. arXiv:0905.0332v1
"... Abstract. We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finitedimensional modules, by directly relating the problem to the classical KazhdanLusztig theory. We further verify a parabolic version ..."
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Cited by 2 (0 self)
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Abstract. We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finitedimensional modules, by directly relating the problem to the classical KazhdanLusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category O of the general linear superalgebra. We also prove the super duality conjecture. 1.
Simplicial Hochschild cochains as an Amitsur complex
, 711
"... It is demonstrated that the cochain complex of relative Hochschild Avalued cochains of a depth two extension A  B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End BAB over the centralizer R = A B with grouplike element 1S, which itself ..."
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It is demonstrated that the cochain complex of relative Hochschild Avalued cochains of a depth two extension A  B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End BAB over the centralizer R = A B with grouplike element 1S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S, S)bicomodule R e. This specializes to finite dimensional algebras, Hseparable extensions and HopfGalois extensions. 2000 MSC: 18G25. 1
RELATIVE HOMOLOGICAL ALGEBRA, EQUIVARIANT DE RHAM COHOMOLOGY AND KOSZUL DUALITY
, 2008
"... Abstract. We describe equivariant de Rham cohomology relative to a general (not necessarily finite dimensional compact) Lie group G in terms of a suitable differential graded Ext defined in terms of the standard construction associated with the monad arising from the assignment to a vector space V o ..."
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Abstract. We describe equivariant de Rham cohomology relative to a general (not necessarily finite dimensional compact) Lie group G in terms of a suitable differential graded Ext defined in terms of the standard construction associated with the monad arising from the assignment to a vector space V of the Vvalued de Rham complex of G. We develop a corresponding infinitesimal equivariant cohomology as the relative differential Ext with respect to the cone Cg on the Lie algebra g of G relative to g itself. Appropriate models for the differential graded Ext involving a comparison between the simplicial Weil coalgebra and the Weil coalgebra dual to the familiar ordinary Weil algebra yield small models for equivariant de Rham cohomology including the familiar Weil and Cartan models for the special case where the group is compact and connected. We explain how Koszul duality in de Rham theory results immediately
A primer on computational group homology and cohomology using GAP and SAGE ∗
, 2009
"... These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (noncommu ..."
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These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (noncommutative) ring, what Z[G] is (where G is a group written multiplicatively and Z denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference or proof of everything. Several computer algebra packages are used to illustrate the computations, though for various reasons we have focused on the free, open source packages, such as GAP [Gap] and SAGE [St] (which includes GAP). In particular, Graham Ellis generously allowed extensive use of his HAP [E1] documentation (which is sometimes copied almost verbatim) in the presentation below. Some interesting work not included in this (incomplete) survey is (for example) that of Marcus