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A DIAGONAL ON THE ASSOCIAHEDRA
, 2000
"... To Jim Stasheff on the occasion of his 65th birthday ..."
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To Jim Stasheff on the occasion of his 65th birthday
L∞ algebras and their cohomology
, 1995
"... Abstract. An associative multiplication structure is nothing but a special type of odd codifferential on the tensor coalgebra of a vector space, and an A∞ algebra is simply a more general type of codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an A ∞ ..."
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Abstract. An associative multiplication structure is nothing but a special type of odd codifferential on the tensor coalgebra of a vector space, and an A∞ algebra is simply a more general type of codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an A ∞ algebra, and cyclic cohomology in the presence of an invariant inner product classifies the deformations of the associative algebra into an A ∞ algebra preserving the inner product. Similarly, a Lie algebra or superalgebra structure is simply a special case of an odd codifferential on the exterior coalgebra of a vector space, and a L∞ algebra is given by an arbitrary codifferential. We define ordinary and cyclic cohomology of L ∞ algebras. Lie algebra cohomology classifies deformations of a Lie algebra into an L ∞ algebra. and cyclic cohomology classifies the deformations of the Lie algebra into an L ∞ algebra preserving an invariant inner product. In the case of Lie algebras, the exterior coalgebra is dual to the symmetric coalgebra of the parity reversion of the vector space, and by use of the parity reversion one obtains a realization of the differential in terms of a structure of a differential graded Lie algebra on the space of cochains. 1.
The biderivative and A∞bialgebras
 Homology Homotopy Appl
"... Abstract. An A∞bialgebra is a DGM H equipped with structurally compatible operations { ω j,i: H ⊗i → H ⊗j} such that ( H, ω 1,i) is an A∞algebra and ( H, ω j,1) is an A∞coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products ..."
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Abstract. An A∞bialgebra is a DGM H equipped with structurally compatible operations { ω j,i: H ⊗i → H ⊗j} such that ( H, ω 1,i) is an A∞algebra and ( H, ω j,1) is an A∞coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End(TH). To Jim Stasheff on the occasion of his 68th birthday. 1.
HIGHER HOMOTOPY HOPF ALGEBRAS FOUND: A 10 YEAR RETROSPECTIVE
, 709
"... Abstract. The search for “higher homotopy Hopf algebras ” (known today as A∞bialgebras) was initiated by this author in a talk at Jim Stasheff’s 1996 schriftfest entitled “In Search of Higher Homotopy Hopf Algebras. ” The idea in that talk was to think of a DG bialgebra as some (unknown) higher hom ..."
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Abstract. The search for “higher homotopy Hopf algebras ” (known today as A∞bialgebras) was initiated by this author in a talk at Jim Stasheff’s 1996 schriftfest entitled “In Search of Higher Homotopy Hopf Algebras. ” The idea in that talk was to think of a DG bialgebra as some (unknown) higher homotopy structure with trivial higher order structure and apply a graded version of Gerstenhaber and Schack’s bialgebra deformation theory. In retrospect, the bi(co)module structure encoded in the differential detects some (but not all) of the A∞bialgebra structure relations; we refer to such deformations as quasiA∞bialgebras. This motivated the discovery of A∞bialgebras given by S. Saneblidze and this author in 2005. 1.
INFINITY ALGEBRAS, COHOMOLOGY AND CYCLIC COHOMOLOGY, AND INFINITESIMAL DEFORMATIONS
, 2001
"... Abstract. An A ∞ algebra is given by a codifferential on the tensor coalgebra of a (graded) vector space. An associative algebra is a special case of an A∞ algebra, determined by a quadratic codifferential. The notions of Hochschild and cyclic cohomology generalize from associative to A ∞ algebras, ..."
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Abstract. An A ∞ algebra is given by a codifferential on the tensor coalgebra of a (graded) vector space. An associative algebra is a special case of an A∞ algebra, determined by a quadratic codifferential. The notions of Hochschild and cyclic cohomology generalize from associative to A ∞ algebras, and classify the infinitesimal deformations of the algebra, and those deformations preserving an invariant inner product, respectively. Similarly, an L ∞ algebra is given by a codifferential on the exterior coalgebra of a vector space, with Lie algebras being special cases given by quadratic codifferentials. There are natural definitions of cohomology and cyclic cohomology, generalizing the usual Lie algebra cohomology and cyclic cohomology, which classify deformations of the algebra and those which preserve an invariant inner product. This article explores the definitions of these infinity algebras, their cohomology and cyclic cohomology, and the relation to their infinitesimal deformations. 1.
Obstructions to Deformations of D.G. Modules
, 1995
"... Let k be a field and n ≥ 1. There exists a differential graded kmodule (V,d) and various approximations to a differential d + td1 + t 2 d2 + · · · + t n dn on V [[t]], one of which is a nontrivial polynomial deformation, another is obstructed, and another is unobstructed at order n. The analogou ..."
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Let k be a field and n ≥ 1. There exists a differential graded kmodule (V,d) and various approximations to a differential d + td1 + t 2 d2 + · · · + t n dn on V [[t]], one of which is a nontrivial polynomial deformation, another is obstructed, and another is unobstructed at order n. The analogous problem in the category of kalgebras in characteristic zero remains a longstanding open question. 1
INFINITY ALGEBRAS, MASSEY PRODUCTS, AND DEFORMATIONS
, 1998
"... The notion of a Massey Fproduct was introduced in [4] in order to give a unified description of various deformation problems in terms of an extension of the usual Massey product. The usual Massey product which arises in algebra deformation theory describes the conditions under which an infinitesima ..."
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The notion of a Massey Fproduct was introduced in [4] in order to give a unified description of various deformation problems in terms of an extension of the usual Massey product. The usual Massey product which arises in algebra deformation theory describes the conditions under which an infinitesimal deformation can be