35,353 research outputs found

    Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials

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    We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in R2R^2. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on R2R^2.Comment: 33 pages, plain TeX. To apper in Phil. Trans. London Math. Soc. Please typeset only the file rf.te

    Divergence operators and odd Poisson brackets

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    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", Δ\Delta, of Batalin-Vilkovisky 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 Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).Comment: 27 pages; new Section 1, introduction and conclusion re-written, typos correcte

    On the algebra of cornered Floer homology

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    Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary an algebra-module over this 2-algebra, such that a natural gluing property is satisfied. Moreover, with a view toward the structure of a potential Floer homology theory of 3-manifolds with codimension-two corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.Comment: a few minor revision

    One class of wild but brick-tame matrix problems

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    We present a class of wild matrix problems (representations of boxes), which are "brick-tame," i.e. only have one-parameter families of \emph{bricks} (representations with trivial endomorphism algebra). This class includes several boxes that arise in study of simple vector bundles on degenerations of elliptic curves, as well as those arising from the coadjoint action of some linear groups.Comment: 19 page

    Symmetries of modules of differential operators

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    Let F_λ(S1){\cal F}\_\lambda(S^1) be the space of tensor densities of degree (or weight) λ\lambda on the circle S1S^1. The space Dk_λ,μ(S1){\cal D}^k\_{\lambda,\mu}(S^1) of kk-th order linear differential operators from F_λ(S1){\cal F}\_\lambda(S^1) to F_μ(S1){\cal F}\_\mu(S^1) is a natural module over Diff(S1)\mathrm{Diff}(S^1), the diffeomorphism group of S1S^1. We determine the algebra of symmetries of the modules Dk_λ,μ(S1){\cal D}^k\_{\lambda,\mu}(S^1), i.e., the linear maps on Dk_λ,μ(S1){\cal D}^k\_{\lambda,\mu}(S^1) commuting with the Diff(S1)\mathrm{Diff}(S^1)-action. We also solve the same problem in the case of straight line R\mathbb{R} (instead of S1S^1) and compare the results in the compact and non-compact cases.Comment: 29 pages, LaTeX, 4 figure

    Action of a finite quantum group on the algebra of complex NxN matrices

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    Using the fact that the algebra M := M_N(C) of NxN complex matrices can be considered as a reduced quantum plane, and that it is a module algebra for a finite dimensional Hopf algebra quotient H of U_q(sl(2)) when q is a root of unity, we reduce this algebra M of matrices (assuming N odd) into indecomposable modules for H. We also show how the same finite dimensional quantum group acts on the space of generalized differential forms defined as the reduced Wess Zumino complex associated with the algebra M.Comment: 11 pages, LaTeX, uses diagrams.sty, to be published in "Particles, Fields and Gravitation" (Lodz conference), AIP proceeding
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