Results 1  10
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13
Hochschild cohomology and moduli spaces of strongly homotopy associative algebras
 Homology Homotopy Appl
"... Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a twocell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hoc ..."
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Cited by 7 (5 self)
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Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a twocell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A∞algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A∞algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest. Keywords: A∞algebra, derivation, Hochschild cohomology, formal power series. 1.
Cohomology theories for homotopy algebras and noncommutative geometry
, 2007
"... This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞, C∞ and L∞algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodg ..."
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Cited by 5 (3 self)
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This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞, C∞ and L∞algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C∞algebras. This generalizes and puts in a conceptual framework previous work by Loday and GerstenhaberSchack.
On PMinimal Homological Models of Twisted Tensor Products of Elementary Complexes Localised over a Prime
"... In this paper, working over Z (p) and using algebra perturbation results from [18], pminimal homological models of twisted tensor products (TTPs) of Cartan 's elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological m ..."
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Cited by 4 (3 self)
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In this paper, working over Z (p) and using algebra perturbation results from [18], pminimal homological models of twisted tensor products (TTPs) of Cartan 's elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model of a indecomposable pminimal TTP of length # (# 2) of exterior and divided power algebras is a tensor product of kindecomposable (k #) pminimal TTPs of exterior and divided power algebras.
R.: Some naturally occurring examples of A∞bialgebras
"... Abstract. Let p be an odd prime. When n ≥ 3, we show that each tensor factor of form E ⊗ Γ in H ∗ (Z, n; Zp) is an A∞infinity bialgebra with nontrivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E ⊗Γ is a naturally occurring example of a ..."
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Cited by 3 (2 self)
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Abstract. Let p be an odd prime. When n ≥ 3, we show that each tensor factor of form E ⊗ Γ in H ∗ (Z, n; Zp) is an A∞infinity bialgebra with nontrivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E ⊗Γ is a naturally occurring example of an A∞bialgebra whose internal structure is wellunderstood. 1.
On the Classification of Moore Algebras and their Deformations
 Homology, Homotopy and Applications
"... Abstract. In this paper we will study deformations of A∞algebras. We will also answer questions relating to Moore algebras which are one of the simplest nontrivial examples of an A∞algebra. We will compute the Hochschild cohomology of odd Moore algebras and classify them up to a unital weak equiva ..."
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Cited by 2 (2 self)
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Abstract. In this paper we will study deformations of A∞algebras. We will also answer questions relating to Moore algebras which are one of the simplest nontrivial examples of an A∞algebra. We will compute the Hochschild cohomology of odd Moore algebras and classify them up to a unital weak equivalence. We will construct miniversal deformations of particular Moore algebras and relate them to the universal odd and even Moore algebras. Finally we will conclude with an investigation of formal oneparameter deformations of an A∞algebra. 1.
Lazarev; Symplectic A∞algebras and string topology operations
"... Abstract. In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincaré duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introdu ..."
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Cited by 1 (0 self)
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Abstract. In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincaré duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading ‘string topology’. Our method is based on obstruction theory for C∞algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.
UNIVERSAL OPERATIONS IN HOCHSCHILD HOMOLOGY
"... Abstract. We provide a general method for finding all natural operations on the Hochschild complex of Ealgebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A∞algebras. We show that the chain complex of all such nat ..."
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Cited by 1 (1 self)
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Abstract. We provide a general method for finding all natural operations on the Hochschild complex of Ealgebras, where E is any algebraic structure encoded in a PROP with multiplication, as for example the PROP of Frobenius, commutative or A∞algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasiisomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and KontsevichSoibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by TradlerZeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of nontrivial operations and use these to produce nontrivial higher string topology operations, which had so far been elusive.
Homotopy Algebras and . . .
, 2004
"... We study cohomology theories of strongly homotopy algebras, namely A∞, C ∞ and L∞algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞algebras thus generalising previous work by Loday and GerstenhaberSchack. These results are then used to show that a C∞algebr ..."
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We study cohomology theories of strongly homotopy algebras, namely A∞, C ∞ and L∞algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞algebras thus generalising previous work by Loday and GerstenhaberSchack. These results are then used to show that a C∞algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞algebra (an∞generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the ‘string topology ’ operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincaré
The infinite associahedron and R.J. Thompson’s group T
"... In this paper we construct a cellular complex which is an infinite analogue to Stasheff’s associahedra. We prove that it is contractible and state that its (combinatorial) automorphism group is isomorphic to a semidirect product of R.J. Thompson’s group T with Z/2Z. MSC classification: 20F65, 20F38 ..."
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In this paper we construct a cellular complex which is an infinite analogue to Stasheff’s associahedra. We prove that it is contractible and state that its (combinatorial) automorphism group is isomorphic to a semidirect product of R.J. Thompson’s group T with Z/2Z. MSC classification: 20F65, 20F38, 57Q05, 57M07.