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49
Hopf algebras, cyclic cohomology and the transverse index theorem
 Comm. Math. Phys
, 1998
"... In this paper we present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. ..."
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Cited by 142 (18 self)
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In this paper we present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations.
Hochschild cohomology of the Weyl algebra and traces in deformation quantization
 Duke Math. J
, 2005
"... Abstract. We give a formula for a cocycle generating the Hochschild cohomology of the Weyl algebra with coefficients in its dual. It is given by an integral over the configuration space of ordered points on a circle. Using this formula and a noncommutative version of formal geometry, we obtain an e ..."
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Cited by 15 (3 self)
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Abstract. We give a formula for a cocycle generating the Hochschild cohomology of the Weyl algebra with coefficients in its dual. It is given by an integral over the configuration space of ordered points on a circle. Using this formula and a noncommutative version of formal geometry, we obtain an explicit expression for the canonical trace in deformation quantization of symplectic manifolds. 1.
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.
LEIBNIZ COHOMOLOGY FOR DIFFERENTIABLE MANIFOLDS
"... The goal of this paper is to extend Loday’s Leibniz cohomology [L,P] from a Lie algebra invariant to an invariant for differentiable manifolds so that Leibniz cohomology is a noncommutative version of de Rham cohomology. The noncommutativity arises by considering a cochain complex of tensors (from ..."
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Cited by 10 (5 self)
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The goal of this paper is to extend Loday’s Leibniz cohomology [L,P] from a Lie algebra invariant to an invariant for differentiable manifolds so that Leibniz cohomology is a noncommutative version of de Rham cohomology. The noncommutativity arises by considering a cochain complex of tensors (from differential geometry) which are not necessarily skewsymmetric.
M.: General solution of the consistency equation
 Phys. Lett
, 1992
"... We produce the general solution of the WessZumino consistency condition for gauge theories of the Yangmills type, for any ghost number and form degree. We resolve the problem of the cohomological independence of these solutions. In other words we fully describe the local version of the cohomology ..."
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Cited by 6 (0 self)
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We produce the general solution of the WessZumino consistency condition for gauge theories of the Yangmills type, for any ghost number and form degree. We resolve the problem of the cohomological independence of these solutions. In other words we fully describe the local version of the cohomology of the BRS operator, modulo the differential on space–time. This in particular includes the presence of external fields and non–trivial topologies of space–time.
DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS II
, 2007
"... Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartantype: the Contact and the Hamiltonian Lie algebras. 1. ..."
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Cited by 6 (5 self)
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Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartantype: the Contact and the Hamiltonian Lie algebras. 1.
EXTENSIONS OF SUPER LIE ALGEBRAS
"... Abstract. We study (nonabelian) extensions of a given super Lie algebra, identify a cohomological obstruction to the existence, parallel to the known one for Lie algebras. An analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is sp ..."
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Cited by 4 (0 self)
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Abstract. We study (nonabelian) extensions of a given super Lie algebra, identify a cohomological obstruction to the existence, parallel to the known one for Lie algebras. An analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. 1. Introduction. The
Extensions of Lie algebras
"... Abstract. We review (nonabelian) extensions of a given Lie algebra, identify a 3dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. 1. ..."
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Cited by 3 (0 self)
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Abstract. We review (nonabelian) extensions of a given Lie algebra, identify a 3dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. 1. Introduction. The
DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA
, 2007
"... Abstract. We compute the infinitesimal deformations of the restricted Melikian Lie algebra in characteristic 5. 1. ..."
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Cited by 2 (2 self)
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Abstract. We compute the infinitesimal deformations of the restricted Melikian Lie algebra in characteristic 5. 1.
The eigenvalues of the Laplacian for the homology of the Lie algebra corresponding to a poset
, 1995
"... In this paper we study the spectral resolution of the Laplacian L of the Koszul complex of the Lie algebras corresponding to a certain class of posets. Given a poset P on the set f1; 2; . . . ; ng, we define the nilpotent Lie algebra L P to be the span of all elementary matrices z x;y , such that x ..."
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In this paper we study the spectral resolution of the Laplacian L of the Koszul complex of the Lie algebras corresponding to a certain class of posets. Given a poset P on the set f1; 2; . . . ; ng, we define the nilpotent Lie algebra L P to be the span of all elementary matrices z x;y , such that x is less than y in P . In this paper, we make a decisive step toward calculating the Lie algebra homology of L P in the case that the Hasse diagram of P is a rooted tree. We show that the Laplacian L simplifies significantly when the Lie algebra corresponds to a poset whose Hasse diagram is a tree. The main result of this paper determines the spectral resolutions of three commuting linear operators whose sum is the Laplacian L of the Koszul complex of L P in the case that the Hasse diagram is a rooted tree. We show that these eigenvalues are integers, give a combinatorial indexing of these eigenvalues and describe the corresponding eigenspaces in representationtheoretic terms. The homology ...