Automorphisms and forms of lie algebras of cartan type

Let L be any of simple Lie algebras of Cartan type L = X⁽²⁾(m : n : Φ), X = 5, H or K, n≠1, over a field F of characteristic p > 3, and R a commutative ring extension of F. Then AutR(R?L)≌ Aut R(R?21 (m : n) : R?L). It follows that, all forms of L are standard and thereby are determined     

Generalized cartan type k lie algebras in characteristic 0

The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra F[x₀, x₁,…, x_n,x_n?1,…,x_?n], where F is a field of characteristic 0, was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials F[x₀,x₁,…, x_n,x_?1,…,x_?n,X₀ ^?1x₁ ^-1,…,x_n ^?1,…,x_?1 ^?1…,x_?n ^?1]A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, determine all possible     

On simple semiabelian p-adic lie algebras

A quantization of a non-standard rational solution of CYBE for sl ₂ is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian Y{sl ₂). This quantum object (deformed Yangian Y _ηξ( sl ₂)) is a two-parametric deformation of the universal enveloping algebra U(sl ₂[u]) of the polynomial current algebra sl ₂[u]. We consider the pseudotriangular structure on Y _ηξ( sl ₂), the quantum double DY _ηξ(sl ₂), its the universal R-matrix and also the RTT-realization of Y _ηξ( sl ₂).     

Enumeration of maximal subalgebras in free restricted lie algebras

Given a finitely generated restricted Lie algebra L over the finite field \(\mathbb{F}_q \), and n ≥ 0, denote by a _n (L) the number of restricted subalgebras H ? L with \(\dim _{\mathbb{F} _q} \) L/H = n. Denote by ã _n (L) the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra L = F _d of rank d ≥ 2, we find the asymptotics of ã _n (F _d ) and show that it coincides with the asymptotics of a _n (F _d ) which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras H ? F _d of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.