Invariants of Lie algebras extended over commutative algebras without unit

Abstract

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.

Deformations of current Lie algebras. I. Small algebras in characteristic 2

“…There are general formulas for H 2 (L ⊗ A, K) for an arbitrary Lie algebra L (see [Z3,Theorem 1]), but they are valid in characteristic = 2, 3. To extend these results to the case of characteristic 2, new notions and techniques will be needed.…”

Yet another proof of the Ado theorem

“…Additionally, one may try to employ derivations of L ⊗ tK[t]/(t n ) of the form other than id L ⊗D, where D is a derivation of tK[t]/(t n ). As explained in [Zu,§3], the full description of derivations of such current Lie algebras is, probably, a difficult task, but one may try, for example, to employ derivations of the forms listed in [Zu,Theorem 3].…”

Commutative 2-cocycles on Lie algebras

“…This generalizes [DB,Theorem 1.1], where the same result is proved for Lie algebras of classical type by performing computations with the corresponding root system. Another proof for such Lie algebras could be derived by combining results of [L] and [Zu3]. In a sense, root space computations in [DB] are equivalent to the appropriate part of computations in [L].…”

“…It is possible to extend Theorem 4.1 and Corollary 4.2 to various generalizations of current Lie algebras, such as twisted algebras, extended affine Lie algebras, toroidal Lie algebras, Lie algebras graded by root systems, etc. Some of computations could be quite cumbersome, but all of them seem to be amenable to the technique used in [Zu3].…”

“…for x, y ∈ L, a, b ∈ A, d ∈ D, and z belongs to the center. Note that the semidirect sum (L ⊗ A) D is a quotient by the 1-dimensional central ideal Kz, and (L ⊗ A) ⊕ Kz is a subalgebra, the latter being central extension of the current Lie algebra L ⊗ A (for generalities about central extensions of current Lie algebras, see [Zu2,Zu3,NW]). Specializing this construction to the case where K is an algebraically closed field of characteristic zero, L = g, a simple finite-dimensional Lie algebra, •, • is the Killing form on g…”