A periodisation of semisimple Lie algebras

Abstract

In this text we study classical Lie algebras. We prove that a periodisation of such Lie algebras without ¢ ¡£ -component can be presented as a free graded Lie algebra modulo quadratic relations only. Our approach will be through a Chevalley basis and our method relies on elementary tools only.

Commutative 2-cocycles on Lie algebras

“…2 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,Zu3]. In a sense, root space computations in [DB] are equivalent to the appropriate part of computations in [L].…”

Invariants of Lie algebras extended over commutative algebras without unit

“…In [La1], Anna Larsson studied periodization of semisimple finite-dimensional Lie algebras g over any field K of characteristic 0. She proved that, unless g contains direct summands isomorphic to sl(2), its periodization possesses a presentation with only quadratic relations.…”

“…We do not touch upon superalgebras here, and it seems to be an interesting task to tackle the results and questions from [La2] from this paper's viewpoint. Interest in periodizations of Lie superalgebras, and whether they admit generators subject to quadratic relations, arose from an earlier work of Löfwall and Roos [LR] about some amazing Hopf algebras (for further details, see [La1] and [La2]).…”

“…However, there are many interesting examples of current algebras where A is not unital. For example, in [La1], the so-called periodization of semisimple Lie algebras g was considered, which is nothing but g ⊗ tC [t]. It is known that the second homology of any nilpotent Lie algebra with trivial coefficients has interpretation in terms of presentation of the algebra, so allowing A to be nilpotent allows us to obtain presentation of L ⊗ A irrespective of the properties of L.…”

“…In § §1-3 we establish the general formulae for 2-cocycles, symmetric invariant bilinear forms, and get partial results about derivations of the current Lie algebras, respectively. This is followed by applications: in §4 we reprove the result from [La1] about presentations of periodizations of the semisimple Lie algebras. In passing, we also mention how to derive from our results the theorem from [Be] about semisimple components of the derivation algebras of certain current Lie algebras, and the Serre defining relations between Chevalley generators of the non-twisted affine Kac-Moody algebra. In all these cases, the absence of unit in A is essential.…”