Realization of graded-simple algebras as loop algebras

Abstract

Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize a‰ne Kac-Moody Lie algebras. In this paper, we obtain necessary and su‰cient conditions for a Z n -graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and su‰cient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group.We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields.

Graded simple Lie algebras and graded simple representations

“…In Sect. 5, for arbitrary additive abelian group Q, using Correspondence Theorem 7.1.1 from [1], we prove that any Q-graded simple Lie algebra g over an algebraically closed field k with dim g < |k| has to be of the form g(Q, P, a), for some simple Lie algebra a with a Q/P-grading, see Theorem 15.…”

“…For a given abelian group Q, a classification of Q-gradings (up to isomorphism) on classical simple Lie algebras over an algebraically closed field of characteristic different from 2 was obtained in [2,5], see also [7]. In [1], one finds some characterizations of graded-central-simple algebras with split centroid, see Correspondence Theorem 7.1.1 in [1]. In general, it is difficult to find the centroid for a graded simple algebra.…”

Classification of involutions on graded-division simple real algebras

“…Hence, D with its G-grading can be obtained from a simple algebra with a grading by the quotient group G/ f by means of the loop construction (see [1]): (1-e) and (1-g) from (1-a); (1-f) and (1-h) from (1-b); and (1-i) from either (1-a) or (1-b).…”

“…Fix an isomorphism D ∼ = End C (V ) and a hermitian form h on V that defines ϕ, that is, ϕ = σ h as in Equation (1). For any X ∈ D, set h X (v, w) := h(v, Xw) for all v, w ∈ V .…”

“…For (2), it is necessary and sufficient to prove that X t X s X t = X st 2 . Indeed, pick u ∈ T such that u 2 = s and compute: X t X s X t = β(t, s)X s X 2 t = β(t, s)X s X t 2 = β(t, s)β(u, t) 2 X st 2 = X st 2 , where in the second last step we have used (1).…”