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85
Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations
 Euromath Bull
, 1999
"... A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are ..."
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Cited by 41 (3 self)
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A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are indicated.
Classification of finitedimensional triangular Hopf algebras with the Chevalley property
 Mathematical Research Letters
"... Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose ..."
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Cited by 18 (9 self)
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Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.
Homogeneous algebras, statistics and combinatorics
"... After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: ..."
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Cited by 17 (9 self)
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After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one called the parafermionic (parabosonic) algebra is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with D degrees of freedom while the second is the plactic algebra that is the algebra of the plactic monoid with entries in {1, 2,..., D}. In the case D = 2 we describe the relations with the cubic ArtinSchelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D = 2 extends as an action of the quantum group GLp,q(2) on the generic cubic ArtinSchelter regular algebra of type S1; p and q being related to the ArtinSchelter parameters. It is claimed that this has a counterpart for any integer D ≥ 2.
Divergence operators and odd Poisson brackets
 Ann. Inst. Fourier
"... Abstract. We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, ∆, of Batalin ..."
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Cited by 16 (1 self)
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Abstract. We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, ∆, of BatalinVilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the KoszulSchouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).
Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebra
 J. Pure Appl. Algebra
"... We propose a detailed systematic study of a group H2 L (A) associated, by elementary means of lazy 2cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize G.I. Kac’s exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: ..."
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Cited by 13 (3 self)
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We propose a detailed systematic study of a group H2 L (A) associated, by elementary means of lazy 2cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize G.I. Kac’s exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a KacSchauenburgtype sequence for double crossed products of possibly infinitedimensional Hopf algebras. Finally the explicit computation of H2 L (A) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed. Key words: Hopf 2cocycle, Galois objects, biGalois objects.
A Selective History of the Stonevon Neumann Theorem
 CONTEMPORARY MATHEMATICS
"... The names of Stone and von Neumann are intertwined in what is now known as the Stonevon Neumann Theorem. We discuss the origins of this theorem, the contributions to it of Stone and von Neumann, the ways the theorem has been reformulated, and some of the varied mathematics that has grown out of it ..."
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Cited by 13 (0 self)
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The names of Stone and von Neumann are intertwined in what is now known as the Stonevon Neumann Theorem. We discuss the origins of this theorem, the contributions to it of Stone and von Neumann, the ways the theorem has been reformulated, and some of the varied mathematics that has grown out of it. At the end we discuss certain generalizations or analogues of the Stonevon Neumann Theorem which are still subjects of current research, such as a new C∗algebra attached to the canonical commutation relations of quantum field theory, and supersymmetric versions of the theorem.
Kontsevich quantization and invariant distributions on Lie groups, preprint math.QA/9910104
"... We study Kontsevich’s deformation quantization for the dual of a finitedimensional real Lie algebra (or superalgebra) g. In this case the Kontsevich ⋆product defines a new convolution on S(g), regarded as the space of distributions supported at 0 ∈ g. For p ∈ S(g), we show that the convolution ope ..."
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Cited by 11 (2 self)
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We study Kontsevich’s deformation quantization for the dual of a finitedimensional real Lie algebra (or superalgebra) g. In this case the Kontsevich ⋆product defines a new convolution on S(g), regarded as the space of distributions supported at 0 ∈ g. For p ∈ S(g), we show that the convolution operator f ↦− → p ⋆ f is a differential operator with analytic germ. We use this fact to prove a conjecture of Kashiwara and Vergne on invariant distributions on a Lie group G. This yields a new proof of Duflo’s result on local solvability of biinvariant differential operators on a Lie group G. Moreover, this new proof extends to Lie supergroups. 0
Hamiltonian versus Lagrangian formulation of supermechanics
 J. Phys. A: Math. Gen
, 1997
"... We take advantage of different generalizations of the tangent manifold to the context of graded manifolds, together with the notion of super section along a morphism of graded manifolds, to obtain intrinsic definitions of the main objects in supermechanics such as, the vertical endomorphism, the can ..."
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Cited by 10 (3 self)
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We take advantage of different generalizations of the tangent manifold to the context of graded manifolds, together with the notion of super section along a morphism of graded manifolds, to obtain intrinsic definitions of the main objects in supermechanics such as, the vertical endomorphism, the canonical and the Cartan’s graded forms, the total time derivative operator and the super–Legendre transformation. In this way, we obtain a correspondence between the Lagrangian and the Hamiltonian formulations of supermechanics. Keywords: Tangent superbundle, Supervector field along a morphism, Super–Legendre transformation.
lectures on Jet manifolds in classical and quantum field theory”, eprint, arXiv: mathph/0203040
"... Abstract. These Lectures summarize the relevant material on existent applications of jet manifold techniques to classical and quantum field theory. The following topics are included: 1. Fibre bundles, 2. Jet manifolds, 3. Connections, 4. Lagrangian field theory, 5. Gauge theory of principal connecti ..."
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Cited by 10 (1 self)
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Abstract. These Lectures summarize the relevant material on existent applications of jet manifold techniques to classical and quantum field theory. The following topics are included: 1. Fibre bundles, 2. Jet manifolds, 3. Connections, 4. Lagrangian field theory, 5. Gauge theory of principal connections, 6. Higher order jets, 7. Infinite order jets, 8. The variational bicomplex, 9. Geometry of simple graded manifolds, 10. Jets of ghosts and antifields.
A geometrical version of Noether's theorem in supermechanics.
 Math. Phys
, 1994
"... this paper, the notion of supervector field along a morphism of supermanifolds is introduced and is used to obtain a version of Noether's theorem admitting a converse, that is generalizable to higher order supermechanics. Keywords: Supervector field along a morphism, Noether's theorem. ..."
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Cited by 8 (5 self)
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this paper, the notion of supervector field along a morphism of supermanifolds is introduced and is used to obtain a version of Noether's theorem admitting a converse, that is generalizable to higher order supermechanics. Keywords: Supervector field along a morphism, Noether's theorem.