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285
PoincaréBirkhoffWitt theorem for quadratic algebras of Koszul type
 Zbl 0860.17002 MR 1383469
, 1996
"... 0.1. Homogeneous quadratic algebras. Let V be a vector space over some field k and let T(V) = ⊕ T i be its tensor algebra over k. Fix a subspace R ⊂ T 2 = V ⊗ V, consider the twosided ideal J(R) in T(V) generated by R and denote by Q(V, R) the quotient algebra T(V)/J(R). This is what is known as ..."
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Cited by 54 (1 self)
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0.1. Homogeneous quadratic algebras. Let V be a vector space over some field k and let T(V) = ⊕ T i be its tensor algebra over k. Fix a subspace R ⊂ T 2 = V ⊗ V, consider the twosided ideal J(R) in T(V) generated by R and denote by Q(V, R) the quotient algebra T(V)/J(R). This is what is known as (a homogeneous) quadratic algebra.
Differential invariants and curved BernsteinGelfandGelfand sequences
 Jour. Reine Angew. Math
"... Abstract. We give a simple construction of the BernsteinGelfandGelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential “cup product ” on this sequence, satisfying a Le ..."
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Cited by 45 (2 self)
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Abstract. We give a simple construction of the BernsteinGelfandGelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential “cup product ” on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a widereaching generalization of helicity raising and lowering for conformally invariant field equations.
Deformation Theory And The BatalinVilkovisky Master Equation
 of the Batalin–Vilkovisky approach,” Secondary Calculus and Cohomological Physics (Moscow, 1997), Contemp. Math. 219, AMS
, 1996
"... The BatalinVilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the BatalinVilkovisky approach is here translated into the language of deformation theory. ..."
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Cited by 40 (0 self)
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The BatalinVilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the BatalinVilkovisky approach is here translated into the language of deformation theory. The following exposition is based in large part on work by Marc Henneaux (Bruxelles) especially and with Glenn Barnich (Penn State and Bruxelles) and Tom Lada and Ron Fulp of NCSU (The NonCommutative State University). The first statement of the relevance of deformation theory to the construction of interactive Lagrangians, that I am aware of, is due to Barnich and Henneaux [3]: We point out that this problem can be economically reformulated as a deformation problem in the sense of deformation theory [13], namely that of deforming consistently the master equation. The `ghosts' introduced by Fade'ev and Popov [12] were soon incorporated into the BRSTcohomology approach [7] to a variety ...
The (secret?) homological algebra of the BatalinVilkovisky approach
"... Stasheff ..."
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Variations on deformation quantization
, 2000
"... I was asked by the organisers to present some aspects of Deformation Quantization. Moshé has pursued, for more than 25 years, a research program based on the idea that physics progresses in stages, and one goes from one level of the theory to the next one by a deformation, in the mathematical sense ..."
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Cited by 24 (0 self)
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I was asked by the organisers to present some aspects of Deformation Quantization. Moshé has pursued, for more than 25 years, a research program based on the idea that physics progresses in stages, and one goes from one level of the theory to the next one by a deformation, in the mathematical sense of the word, to be defined in an appropriate category. His study of deformation theory applied to mechanics started in 1974 and led to spectacular developments with the deformation quantization programme. I first met Moshé at a conference in Liège in 1977. A few months later he became my thesis “codirecteur”. Since then he has been one of my closest friends, present at all stages of my personal and mathematical life. I miss him.... I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness –up to equivalence – of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization.
G.J.: Commutative quantum operator algebras
 J. Pure Appl. Algebra
"... ABSTRACT. A key notion bridging the gap between quantum operator algebras [22] and vertex operator algebras [4][8] is the definition of the commutativity of a pair of quantum operators (see section 2 below). This is not commutativity in any ordinary sense, but it is clearly the correct generalizatio ..."
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Cited by 24 (9 self)
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ABSTRACT. A key notion bridging the gap between quantum operator algebras [22] and vertex operator algebras [4][8] is the definition of the commutativity of a pair of quantum operators (see section 2 below). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. The main purpose of the current paper is to begin laying the foundations for a complete mathematical theory of commutative quantum operator algebras. We give proofs of most of the relevant results announced in [22], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators. We dedicate this paper to the memory of Feza Gürsey. 1
Construction of miniversal deformations of Lie algebras
 J. Func. Anal
, 1999
"... Múzeum krt. 68 ..."
The Hidden Group Structure Of Quantum Groups: Strong Duality, Rigidity And Preferred Deformations.
, 1993
"... : A notion of wellbehaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeldtype and their duals, algebras of coefficients of compact semisimple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantu ..."
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Cited by 22 (7 self)
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: A notion of wellbehaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeldtype and their duals, algebras of coefficients of compact semisimple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyalproductlike deformations are naturally found for all FRTmodels on coefficients and C 1 functions. Strong rigidity (H 2 bi = f0g) under deformations in the category of bialgebras is proved and consequences are deduced. AMS classification: Primary 17B37, 16W30, 22C05 46H99, 81R50. Running title : Topological quantum groups. (In press in Communications in Mathematical Physics, end of 1993) 1 Universit'e de Bourgogne  Laboratoire de Physique Math'ematique B.P. 138, 21004 DIJON Cedex  FRANCE, email: flato@satie.ubourgogne.fr 2 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 191046395 U.S.A. email: mgersten@mail.sas.upenn.edu and murray@math...
Natural star products on symplectic manifolds and quantum moment maps
, 2003
"... We define a natural class of star products: those which are given by a series of bidi#erential operators which at order k in the deformation parameter have at most k derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We p ..."
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Cited by 22 (3 self)
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We define a natural class of star products: those which are given by a series of bidi#erential operators which at order k in the deformation parameter have at most k derivatives in each argument. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance and give necessary and sufficient conditions for them to yield a quantum moment map. We show that Kravchenko's su#cient condition [18] for a moment map for a Fedosov star product is also necessary.
Irreducible components of varieties of modules
 J. Reine Angew. Math
"... Abstract. We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory. ..."
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Cited by 20 (4 self)
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Abstract. We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory. 1.