ABSTRACT
Throughout this note k will denote an algebraically closed field of characteristic 01. The unadorned symbol ⊗ will always stand for tensor product over our base field k. We fix once and for all a choice of compatible primitive n-th roots of unity
(ζn) in k×. Thus, ζeen = ζn for all n, e ≥ 1. Our primary objects of study, loop algebras, are constructed out of a pair (A, σ) consisting of a k-algebra A together with an automorphism σ of A that is of finite order (henceforth denoted by m). Before defining loop algebras, let us introduce three rings R ⊂ Sm ⊂ Ŝ that will play an important role in their study. They are
R = k[t±1], Sm = k[t±1/m], and Ŝ = lim−→Sm. This last ring has a very simple interpretation. As a space, Ŝ can (and
will) be naturally identified with ⊕q∈Qktq. The multiplication is then given by bilinear extension of tptq = tp+q. Let then A be a k-algebra. At this point we do not put any assumption on
the nature of A (in fact A may end up being something more general than an algebra as we shall see in the superconformal case later). Let σ ∈ Autk A
be of finite order m. We have the eigenspace decomposition
A = ⊕m−1i=0 Ai¯, Ai¯ := {a ∈ A |σ(a) = ζima} where − : Z→ Zm := Z/mZ is the canonical map. We then define the loop algebra of (A, σ) by
(1.1) L(A, σ) := ⊕i∈ZAi¯ ⊗ ti/m ⊂ A⊗ Sm ⊂ A⊗ Ŝ. The loop algebra has a natural k-algebra structure (infinite dimensional
whenever A 6= 0). Of course, if A is one of the “usual” type of algebras (e.g., associate, Lie, Jordan,...), then so is L(A, σ).
