58 research outputs found
Regular derivations of truncated polynomial rings
Let be an algebraically closed field of characteristic . Let
, a truncated
polynomial ring in variables, and denote by the derivation
algebra of . It is known that the ring of all polynomial
functions on invariant under the action of the group of
is freely generated by elements. Furthermore,
the related quotient morphism is faithfully flat and all its fibres are
irreducible complete intersections. An element is called
if the centraliser of in has the smallest
possible dimension. In this preprint we give an explicit description of regular
elements of and show that a precise analogue of Kostant's
differential criterion for regularity holds in . We also show that
a fibre of the above mentioned quotient morphism is normal if and only if it
consists of regular semisimple elements of .Comment: Many typos and a serious error in the proof of Theorem 3 are
correcte
Primitive ideals, non-restricted representations and finite W-algebras
We prove that all finite W-algebras associated with nilpotent elements e in a
complex semisimple Lie algebra g have finite-dimensional representations. In
order to obtain this result we establish a connection between primitive ideals
of U(g) attached to the nilpotent orbit containing e and finite-dimensional
representations of the reduced enveloping algebra assiciated with e over an
algebraically closed field of finite characteristic.Comment: 19 pages, minor corrections and up-dates, the new version is accepted
for publicatio
Multiplicity-free primitive ideals associated with rigid nilpotent orbits
We prove that any finite W-algebra U(g,e) admits a one-dimensional
representation fixed by the action of the component group of the centraliser of
e. As a consequence, for any nilpotent orbit O in g there exists a
multiplicity-free (and hence completely prime) primitive ideal of the universal
enveloping algebra U(g) whose associated variety coincides with the Zariski
closure of O.Comment: minor changes and corrections; the present version has been accepted
for publicatio
Enveloping algebras of Slodowy slices and Goldie rank
It is known that any primitive ideal I of U(g) whose associated variety
contains a nilpotent element e in its open G-orbit admits a finite generalised
Gelfand-Graev model which is a finite dimensional irreducible module over the
finite W-algebra U(g,e). We prove that if V is such a model for I, then the
Goldie rank of the primitive quotient U(g)/I always divides the dimension of V.
For g=sl(n), we use a result of Joseph to show that the Goldie rank of U(g)/I
equals the dimension of V and we show that the equality conntinues to hold
outside type A provided that the Goldie field of U(g)/I is isomorphic to a Weyl
skew-field. As an application of this result, we disprove Joseph's conjecture
on the structure of the Goldie fields of primitive quotients of U(g) formulated
in the mid-70s.Comment: This version is accepted for publication (many typos and minor errors
are corrected, several new remarks, etc
Classification of finite dimensional simple Lie algebras in prime characteristics
We give a comprehensive survey of the theory of finite dimensional Lie
algebras over an algebraically closed field of characteristic p>0 and announce
that for p>3 the classification of finite dimensional simple Lie algebras is
complete. Any such Lie algebra is up to isomorphism either classical (i.e.
comes from characteristic 0) or a filtered Lie algebra of Cartan type or a
Melikian algebra of characteristic 5.Comment: Revised version: a list of open problems has been added as suggested
by the refere
Simple Lie algebras of small characteristic VI. Completion of the classification
Let L be a finite-dimensional simple Lie algebra over an algebraically closed
field of F characteristic p>3. We prove that if the p-envelope of L in the
derivation algebra of L contains nonstandard tori of maximal dimension, then
p=5 and L is isomorphic to one of the Melikian algebras. Together with our
earlier results this implies that any finite-dimensional simple Lie algebra
over F is of classical, Cartan or Melikian type.Comment: Many typos corrected and introduction extended; the new version is
accepted for publicatio
Quantizing Mishchenko-Fomenko subalgebras for centralizers via affine W-algebras
We use affine W-algebras to quantize Mishchenko-Fomenko subalgebras for
centralizers of nilpotent elements in simple Lie algebras under certain
assumptions that are satisfied for all cases in type A and all minimal
nilpotent cases outside type .Comment: The main result is stated and proved in greater generality. The
present version is accepted for publicatio
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