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HOW TO QUANTIZE ANTIBRACKET
, 2000
"... The uniqueness of (the class of) deformation of Poisson Lie algebra po(2n) has long been a completely accepted folklore. Actually it is wrong as stated, because its validity depends on the class of functions that generate po(2n) (it is true for polynomials but false for Laurent polynomials). We sh ..."
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Cited by 15 (10 self)
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The uniqueness of (the class of) deformation of Poisson Lie algebra po(2n) has long been a completely accepted folklore. Actually it is wrong as stated, because its validity depends on the class of functions that generate po(2n) (it is true for polynomials but false for Laurent polynomials). We show that unlike po(2nm), its quotient modulo center, the Lie superalgebra h(2nm) of Hamiltonian vector fields with polynomial coefficients, has exceptional extra deformations for (2nm) = (22) and only for this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). It turns out that the representation of the deform (after quantization) of the Poisson algebra in the Fock space coincides with the simplest space on which the Lie algebra of commutation relations acts. This coincidence is not necessary for Lie superalgebras.
STRUCTURES OF G(2) TYPE AND NONINTEGRABLE DISTRIBUTIONS IN CHARACTERISTIC p
"... Structures of G(2) type and nonholonomic distributions in characteristic p by ..."
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Cited by 14 (9 self)
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Structures of G(2) type and nonholonomic distributions in characteristic p by
The Howe duality and Lie superalgebras
 Noncommutative Structures in Mathematics and Physics Proceedings of the NATO Advanced Research Workshop
"... Abstract. Howe’s duality is considered from a unifying point of view based on Lie superalgebras. New examples are offered. In particualr, we construct several simplest spinoroscillator representations and compute their highest weights for the “stringy ” Lie superalgebras (i.e., Lie superalgebras of ..."
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Cited by 9 (4 self)
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Abstract. Howe’s duality is considered from a unifying point of view based on Lie superalgebras. New examples are offered. In particualr, we construct several simplest spinoroscillator representations and compute their highest weights for the “stringy ” Lie superalgebras (i.e., Lie superalgebras of complex vector fields (or their nontrivial central extensions) on the supercircle S 1n and its twosheeted cover associated with the Möbius bundle). In our two lectures we briefly review, on the most elementary level, several results and problems unified by “Howe’s duality”. Details will be given elsewhere. The ground field in the lectures is C. In his famous preprint [24] R. Howe gave an inspiring explanation of what can be “dug out ” from H. Weyl’s “wonderful and terrible ” book [55], at least as far as invariant theory is concerned, from a certain unifying viewpoint. According to Howe, much is based on a remarkable correspondence between certain irreducible representations of Lie subalgebras Γ
Orthogonal Polynomials of Discrete Variable and Lie Algebras of Complex Size Matrices
 in Procedings of M Saveliev Memorial Conference, MPI
, 1999
"... Abstract. We give a uniform interpretation of the classical continuous Chebyshev’s and Hahn’s orthogonal polynomials of discrete variable in terms of Feigin’s Lie algebra gl(λ) for λ ∈ C. One can similarly interpret Chebyshev’s and Hahn’s qpolynomials and introduce orthogonal polynomials correspond ..."
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Cited by 9 (5 self)
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Abstract. We give a uniform interpretation of the classical continuous Chebyshev’s and Hahn’s orthogonal polynomials of discrete variable in terms of Feigin’s Lie algebra gl(λ) for λ ∈ C. One can similarly interpret Chebyshev’s and Hahn’s qpolynomials and introduce orthogonal polynomials corresponding to Lie superalgebras. We also describe the real forms of gl(λ), quasifinite modules over gl(λ), and conditions for unitarity of the quasifinite modules. Analogs of tensors over gl(λ) are also introduced. This is a transcript of the talk at MPI, Bonn in the memory of Misha Saveliev (preprint MPI199944 at www.mpimbonn.mpg.de), see also Theor. Math. Phys., v. 123, no. 2, 2000, 205–236 (Russian), 582–609 (English). It consists of three parts: the description of orthogonal polynomials proper (§1, 8, 9) and some auxiliary results: the description of the generating function of the trace and its analogs (§2) and a description of quasifinite modules over gl(λ) for λ ∈ C. For a continuation, see [S]. For recapitulations, see [NU], [NSU]. 1.0. Definitions. The equation of the form
Leites D., The classification of almost affine (hyperbolic) Lie superalgebras
 J. Nonlinear Math. Phys
"... Abstract. We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine (Kac–Moody) but the Lie (super)algebra determined by any submatrix of A, obtained by strikin ..."
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Cited by 8 (3 self)
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Abstract. We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine (Kac–Moody) but the Lie (super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine (Kac– Moody), and all of its Cartan matrices are almost affine. We list all almost affine Lie superalgebras over complex numbers correcting two earlier claims of classification and make available the list of almost affine Lie algebras obtained by Li Wang Lai. 1.
Classification of simple finite dimensional modular Lie superalgebras with indecomposable Cartan matrix
, 2007
"... Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical hypotheses. Either these Lie superalgebras are simple or the quotients of their derived algebras modulo center are simple. Twelve new exceptional simp ..."
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Cited by 8 (6 self)
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Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical hypotheses. Either these Lie superalgebras are simple or the quotients of their derived algebras modulo center are simple. Twelve new exceptional simple modular Lie superalgebras are discovered.
Divided power (co)homology. Presentations of simple finite dimensional modular Lie superalgebras with Cartan matrix
 Homology, Homotopy Appl
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The Shapovalov Determinant for the Poisson Superalgebras
, 2000
"... Among simple Zgraded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C2: these are the Lie superalgebra k L (16) of vector fields on the (16)dimensional supercircle preserving the contact form, and the ..."
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Among simple Zgraded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C2: these are the Lie superalgebra k L (16) of vector fields on the (16)dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(02k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac–Moody version of sh(02k). Using C2 we compute N. Shapovalov determinant for k L (16) and sh(02k), and for the Poisson superalgebras po(02k) associated with sh(02k). A. Shapovalov described irreducible finite dimensional representations of po(0n) and sh(0n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(02k) and sh(02k).
Leites D, Defining relations of almost affine (hyperbolic) Lie superalgebras
 J. Nonl. Math. Phys
, 2010
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