Quantum Tori and the Structure of Elliptic Quasi-simple Lie Algebras

Abstract

We study and classify those tame irreducible elliptic quasi-simple Lie algebras which are simply laced and of rank l 3. The first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates. When the type is D or E the coordinates are Laurent polynomials in & variables, while for type A the coordinates can be any quantum torus in & variables. The next step is to study the universal central extension as well as the derivation algebra of the core. These are related to the first Connes cyclic homology group of the coordinates. The final step is to use this information to give constructions of Lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras.

Derivations and invariant forms of Jordan and alternative tori

“…In §4 we then prove (1) for all Jordan tori, by making use of their classification ([31]). As already mentioned, (1) has been proven for associative tori in [4] and for nonassociative alternative tori in [5]. Our paper provides a slightly more conceptual proof in the latter case.…”

“…That the derivation algebra is a semidirect product of the ideal of inner derivations and a subalgebra had been proven before in [4] for quantum tori and in [5] for Cayley tori, see section 4.3. One of the novelties of this paper is that we provide a conceptual description of this subalgebra as the algebra of central derivations.…”

“…Moreover, it is proven in the recent paper [2] by Allison and Gao that special classes of Jordan tori enter in the description of the centreless cores of extended affine Lie algebras of reduced non-simply-laced types. In the spirit of the paper [4] by Berman, Gao and Krylyuk on extended affine Lie algebras of type A l , l ≥ 3 (or [5] for type A 2 ) the description (1) is an essential ingredient in the classification of all tame extended affine Lie algebras of type A 1 and other types. Another ingredient in the construction of extended affine Lie algebras of type A 1 is invariant forms.…”

A class of irreducible integrable modules for the extended baby TKK algebra

“…Extended affine Lie algebras (EALAs) were first introduced in [Høegh-Krohn and Torrésani 1990] and studied systematically in [Allison et al 1997;Berman et al 1996]. They are natural generalizations of finite-dimensional simple Lie algebras and affine Kac-Moody algebras.…”

Iterated loop algebras

“…It follows that ᏸ k q , where k q is the algebra presented by the generators x 1 , x 2 subject to the relations (9-5). This algebra k q , which is called the quantum torus determined by the matrix q = 1 ζ ζ −1 1 , has arisen in a number of different contexts; see for example [Magid 1978;McConnell and Pettit 1988;Berman et al 1996;Gao 2000]. Note that by Corollary 6.6, the centroid (= centre) of ᏸ is isomorphic toC(ᏸ) = k[t ±1 1 , t ±1 2 ], where t 1 = z 1 and t 2 = z 2 .…”