Results 1  10
of
39
Local BRST cohomology in gauge theories
, 2000
"... The general solution of the anomaly consistency condition (WessZumino equation) has been found recently for YangMills gauge theory. The general form of the counterterms arising in the renormalization of gauge invariant operators (KlubergStern and Zuber conjecture) and in gauge theories of the Yan ..."
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The general solution of the anomaly consistency condition (WessZumino equation) has been found recently for YangMills gauge theory. The general form of the counterterms arising in the renormalization of gauge invariant operators (KlubergStern and Zuber conjecture) and in gauge theories of the YangMills type with non power counting renormalizable couplings has also been worked out in any number of spacetime dimensions. This Physics Report is devoted to reviewing in a selfcontained manner these results and their proofs. This involves computing cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin, with the sources of the BRST variations of the fields (“antifields”) included in the problem. Applications of this computation to other physical questions (classical deformations of the action, conservation laws) are also considered. The general algebraic techniques developed in the Report can be applied to other gauge theories, for which relevant references are given.
A Koszul duality for props
 Trans. of Amer. Math. Soc
"... Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props. ..."
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Cited by 21 (4 self)
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Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.
Operator YangBaxter Equations, Integrable ODEs and Nonassociative Algebras
 J Nonlin Math Phys
"... Abstract. Reductions for systems of ODEs integrable via the standard factorization method (the AdlerKostantSymes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator YangBaxter equations, and some kinds ..."
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Cited by 15 (0 self)
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Abstract. Reductions for systems of ODEs integrable via the standard factorization method (the AdlerKostantSymes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator YangBaxter equations, and some kinds of nonassociative algebras are established. The factorization method (see [1]) is used to integrate a system of ODEs of the form qt = [q+, q], q(0) = q0. (0.1) Here q(t) belongs to a Lie algebra G, which is decomposed (as a vector space) into
Nondegenerate invariant bilinear forms on non–associative algebras, Preprint Freiburg THEP 92/3, to appear
 Acta Math. Univ. Comenianae
"... Abstract. A bilinear form f on a nonassociative algebra A is said to be invariant iff f(ab, c) = f(a, bc) for all a, b, c ∈ A. Finitedimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal st ..."
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Abstract. A bilinear form f on a nonassociative algebra A is said to be invariant iff f(ab, c) = f(a, bc) for all a, b, c ∈ A. Finitedimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal structure of A if f is nondegenerate and introduce the notion of T ∗extension of an arbitrary algebra B (i.e. by its dual space B ∗ ) where the natural pairing gives rise to a nondegenerate invariant symmetric bilinear form on A: = B ⊕ B ∗. The T ∗extension involves the third scalar cohomology H3 (B, K) if B is Lie and the second cyclic cohomology HC 2 (B) if B is associative in a natural way. Moreover, we show that every nilpotent finitedimensional algebra A over an algebraically closed field carrying a nondegenerate invariant symmetric bilinear form is a suitable T ∗extension. As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel’d but not always isomorphic to a Manin triple. Examples involving the Heisenberg and filiform Lie algebras (whose third scalar cohomology is computed) are discussed. 1.
DUALISING COMPLEXES AND TWISTED HOCHSCHILD (CO)HOMOLOGY FOR NOETHERIAN HOPF ALGEBRAS
, 2006
"... Abstract. We show that many noetherian Hopf algebras A have a rigid dualising complex R with R ∼ = ν A 1 [d]. Here, d is the injective dimension of the algebra and ν is a certain kalgebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is her ..."
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Cited by 12 (1 self)
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Abstract. We show that many noetherian Hopf algebras A have a rigid dualising complex R with R ∼ = ν A 1 [d]. Here, d is the injective dimension of the algebra and ν is a certain kalgebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call ν the Nakayama automorphism of A. We prove that ν = S 2 ξ, where S is the antipode of A and ξ is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be nonzero in the top dimension d, when A is an ArtinSchelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincaré duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating ν for A using also the opposite coalgebra structure, we determine a formula for S 4 generalising a 1976 formula of Radford for A finite dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.
The exterior algebra and “spin” of an orthogonal gmodule
 Transform. Groups
"... Let g be a reductive algebraic Lie algebra over an algebraically closed field k of characteristic zero and G is the corresponding connected and simply connected group. The symmetric algebra of a (finitedimensional) gmodule V is the algebra of polynomial ..."
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Cited by 7 (4 self)
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Let g be a reductive algebraic Lie algebra over an algebraically closed field k of characteristic zero and G is the corresponding connected and simply connected group. The symmetric algebra of a (finitedimensional) gmodule V is the algebra of polynomial
Equivariant cohomology and the MaurerCartan equation
, 2005
"... Let G be a compact, connected Lie group, acting smoothly on a manifold M. In their 1998 paper, GoreskyKottwitzMacPherson described a small Cartan model for the equivariant cohomology of M, quasiisomorphic to the standard (large) Cartan complex of equivariant differential forms. In this paper, w ..."
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Let G be a compact, connected Lie group, acting smoothly on a manifold M. In their 1998 paper, GoreskyKottwitzMacPherson described a small Cartan model for the equivariant cohomology of M, quasiisomorphic to the standard (large) Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small Cartan model into the large Cartan model, intertwining the (Sg ∗)invmodule structures and inducing an isomorphism in cohomology. The construction involves the solution of a remarkable inhomogeneous MaurerCartan equation. This solution has further applications to the theory of transgression in the Weil algebra, and to the ChevalleyKoszul theory of the
Graded filiform Lie algebras and symplectic Nilmanifolds
 in “Geometry, Topology, and Mathematical
"... Abstract. We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e1,..., en, [ei, ej] = ci ..."
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Cited by 6 (2 self)
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Abstract. We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e1,..., en, [ei, ej] = cijei+j (Ngraded Lie algebras). In particular we describe the spaces of symplectic cohomology classes for all evendimensional algebras of the list. It is proved that a symplectic filiform Lie algebra g is a filtered deformation of some Ngraded symplectic filiform Lie algebra g0. But this condition is not sufficient. A spectral sequence is constructed in order to answer the question whether a given deformation of a Ngraded symplectic filiform Lie algebra g0 admits a symplectic structure or not. Other applications and examples are discussed.
Origins and breadth of the theory of higher homotopies
, 2007
"... Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least ..."
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Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the AlexanderWhitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative Hspaces, and a careful examination of this extension led Stasheff to the discovery of Anspaces and A∞spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic