212 research outputs found
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
The classification of the simple modular Lie algebras II. The toral structure
AbstractLet L be a simple Lie algebra over an algebraically closed field of characteristic p > 7 and T an optimal torus in some p-envelope Lp. We determine the action of T on the two-sections of L, which have been described in [St4]. We also give some new and noncomputational proofs to determine the conjugacy classes of the tori in W(n; 1) and of the Cartan subalgebras of W(1; n)
Simple Lie algebras of small characteristic IV. Solvable and classical roots
AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic p>3 and T a torus of maximal dimension in the p-envelope of L in DerL. In this paper we describe the T-semisimple quotients of the 2-sections of L relative to T and prove that if all 1-sections of L relative to T are compositionally classical or solvable then L is either classical or a Block algebra or a filtered Lie algebra of type S
Commutative 2-cocycles on Lie algebras
On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear
forms satisfying the usual cocycle equation. We note their relationship with
antiderivations and compute them for some classes of Lie algebras, including
finite-dimensional semisimple, current and Kac-Moody algebras.Comment: v7: minor changes; added ancillary file with GAP cod
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Block algebras with HH1 a simple Lie algebra
The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP). In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules
Lie solvable enveloping algebras of characteristic two
Lie solvable restricted enveloping algebras were characterized by Riley and
Shalev except when the ground field is of characteristic 2. We resolve the
characteristic 2 case here which completes the classification. As an
application of our result, we obtain a characterization of ordinary Lie
algebras over any field whose enveloping algebra is Lie solvable
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