Results 1 
6 of
6
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... ..."
Cohomology of Algebraic Theories
 J. of Algebra
, 1991
"... this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups ..."
Abstract

Cited by 30 (17 self)
 Add to MetaCart
this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups in functor categories; related questions are discussed. Let us recall the denition of the Mac Lane cohomology from [13]. Consider the sets Cn with 2 n elements  ntuples ("1 ; :::; "n ), where " i = 0 or 1, for n > 0 and i 6 n, and the 0tuple ( ) for n = 0. For convenience Cn can be visualized as the set of vertices of an ncube, the product of n copies of the 1cube with vertices 0 and 1. Dene maps 0 i ; 1 i : Cn ! Cn+1 , 1 6 i 6 n + 1, by the equalities 0 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 0; " i+1 ; :::; "n ); 1 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 1; " i+1 ; :::; "n ): For an abelian group A and a set S, let A[S] denote the sum of S copies of the group A. Since the sets Cn are nite, the group A[Cn ] can be identied with the group of all maps t : Cn ! A: Let Q 0 n (A) be the free abelian group generated by the set A[Cn ], i. e., Q 0 n (A) = Z[A[Cn ]]: Following Mac Lane [13], dene for i = 1; 2; :::; n the homomorphisms R i ; S i ; P i : Q 0 n (A) ! Q 0 n 1 (A) by R i = Z[R i ]; S i = Z[S i ]; P i = Z[P i ]; COHOMOLOGY OF ALGEBRAIC THEORIES 257 where R i ; S i ; P i : A[Cn ] ! A[Cn 1 ] are homomorphisms dened for e 2 Cn 1 and t 2 A[Cn ] by (R i t)(e) = t(0 i e); (S i t)(e) = t(1 i e); (P i t)(e) = t(0 i e) + t(1 i e): In [13], Mac Lane denes the boundary homomorphism @ : Q 0 n (A) ! Q 0 n 1 (A) by the equality @ = n X i=1 ( 1) i (P i R i S i ): A generator t : Cn ! A of the group Q 0 n (A) is called a slab when t( ) = 0, for n...
Symmetric bundles and representations of Lie triple systems”, arXiv: math.DG/0710.1543
"... Abstract. We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying reflection space, and we investigate the corresponding forgetf ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing “unusual ” symmetric bundle structures on the tangent bundles of certain symmetric spaces.
Homological Algebra of Racks and Quandles
"... Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expan ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Contents Introduction 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Extensions 5 1.1 Extensions and expansions . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Quandle extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Involutory extensions . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Modules 29 2.1 Rack modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Beck modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 A digression on Gmodules . . . . . . . . . . . . . . . . . . . . . 41 2.4 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 The rack
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
Wolfgang Bertram, Manon Didry 1 Symmetric bundles and representations of Lie triple systems
, 2009
"... Abstract. We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying reflection space, and we investigate the corresponding forgetf ..."
Abstract
 Add to MetaCart
Abstract. We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing “unusual ” symmetric bundle structures on the tangent bundles of certain symmetric spaces.