35,353 research outputs found
Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials
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 . 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 .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
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-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
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
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
Let be the space of tensor densities of degree (or
weight) on the circle . The space of -th order linear differential operators from
to is a natural module over
, the diffeomorphism group of . We determine the
algebra of symmetries of the modules , i.e.,
the linear maps on commuting with the
-action. We also solve the same problem in the case of
straight line (instead of ) 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
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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