Iterated loop algebras

Abstract

Iterated loop algebras are by definition obtained by repeatedly applying the loop construction, familiar from the theory of affine Kac-Moody Lie algebras, to a given base algebra. Our interest in this iterated construction is motivated by its use in the realization of extended affine Lie algebras, but the construction also appears naturally in the study of other classes of algebras. This paper consists of a detailed study of the basic properties of iterated loop algebras.

Galois cohomology and forms of algebras over Laurent polynomial rings

“…Because of (4), the calculations of (5) can be carried on F instead of R. One can now reason as in Proposition 6.3.9 of [24]. By (3) and (4) we have Z/nZ H 2 (R, µ µ µ n ) n Br(R) n Br(F). So for establishing (6), it suffices to show that the element [i] ∈ Z/nZ is represented by the F-algebra A(i, n) ⊗ R F. But this is Proposition 5.7.1 of [24].…”

“…, x n ) is an n-tuple of elements of H. We summarize this by simply saying that "x x x lies in H". 3 Let x x x = (x 1 , x 2 , . .…”

Derivations of multi-loop algebras

“…Since often we consider more than one automorphism of A, we write A s j :¼ A j to emphasize on the role of s. In particular, with this notation, A s 0 means the fixed points of A with respect to the automorphism s. If B is a subalgebra of EndðAÞ such that sBs À1 ¼ B, we denote by s à , the au- [ABP,Example 9.7] that there are iterated loop algebras which are not multi-loop algebras. Since often we consider more than one automorphism of A, we write A s j :¼ A j to emphasize on the role of s. In particular, with this notation, A s 0 means the fixed points of A with respect to the automorphism s. If B is a subalgebra of EndðAÞ such that sBs À1 ¼ B, we denote by s à , the au- [ABP,Example 9.7] that there are iterated loop algebras which are not multi-loop algebras.…”

“…(See [ABP,Example 5.4]. ) In fact using the fact that the s i 's commute, it is easy to see that S p is the Z m p -grading on L pÀ1 induced by the automorphism s p :¼…”

“…For example an iterated loop algebra based on a perfect algebra is perfect, or an iterated loop algebra based on a (perfect) algebra which as a module over its centroid is finitely generated, is finitely generated as a module over its centroid (see [ABP,Theorem 5.5]). Then this allows one to determine which properties of an iterated loop algebra are inherited from its base algebra.…”