ON UNIVERSAL COVERINGS OF LIE TORI

Abstract

Abstract. In this paper we give an introduction to the theory of uni-versal central extensions of perfect Lie algebras. In particular, we willprovide a model for the universal coverings of Lie tori and we show thatautomorphisms and derivations lift to the universal coverings. We alsoprove that the universal covering of a Lie Λ-torus of type ∆ is again a LieΛ-torus of type ∆. 0. IntroductionCentral extensions play an important role in the theory of Lie algebras.Universal central extension of Lie algebras over rings were described in [23, §1],later references on universal central extensions are [12, §1], [15, 1.9], [24, 7.9] or[21]. Central extensions in the category of certain topological Lie algebras arestudied in [17]. Garland studies universal central extensions of Lie algebrasoverfields [12, §1]. In particular, he constructsa model ofuniversalcentralextensionof a perfect Lie algebra, using the universal 2-cocycle, which is different fromthe Van der Kallen , s model (see [23, §1]). Our construction for universal centralextensions of Lie tori is essentially the Van der Kallen’s model.Lie tori play a critical role in the theory of extended affine Lie algebras whichare natural generalization of finite dimensional simple Lie algebras and affineLie algebras. Yoshii and Neher were interested in Lie tori primarily becauseof the connection between Lie tori and extended affine Lie algebras ([26] and[19]). The centerless core of an extended affine Lie algebra is a centerless ofLie torus and conversely any centerless Lie torus is the centerless core of anextended affine Lie algebra [26, Theorem 7.3]. Extended affine Lie algebrasare defined axiomatically by Alison, Azam, Berman, Gao and Pianzola in [1].The various classes of these Lie algebras have been investigated in many papers(see [4, 6, 13, 14, 20, 25, 27] and [16]). Lie tori as well as extended affine Lie