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By Miller G. A.

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But the Lie algebras g⊗tN C[[t]] and h−1 gh, where h runs over those elements of H((t)) for which h(g ⊗ tN C[[t]])h−1 contains n+ , generate the entire Lie algebra g((t)). Therefore v is g((t))-invariant. We conclude that V is a trivial representation of g((t)). Thus, we find that the class of algebraic representations of loop groups turns out to be too restrictive. , the representations of G((t)) considered as a Lie group. But it is easy to see that the result would be the same. Replacing G((t)) by its central extension G would not help us much either: irreducible integrable representations of G are parameterized by dominant integral weights, and there are no extensions between them [K2].

To get closer to answering these questions, we wish to discuss two more steps that we can make in the above discussion to get to the types of categories with an action of the loop group that we will consider in this book. 3 A toy model At this point it is instructive to detour slightly and consider a toy model of our construction. Let G be a split reductive group over Z, and B its Borel subgroup. A natural representation of G(Fq ) is realized in the space of complex- (or Q -) valued functions on the quotient G(Fq )/B(Fq ).

Each point s ∈ S defines an algebra homomorphism 31 32 Vertex algebras (equivalently, a character) ρs : Z(C) → C (evaluation of a function at the point s). We define the full subcategory Cs of C whose objects are the objects of C on which Z(C) acts according to the character ρs . It is instructive to think of the category C as “fibering” over S, with the fibers being the categories Cs . Now suppose that C = A -mod is the category of left modules over an associative C-algebra A. Then A itself, considered as a left A-module, is an object of C, and so we obtain a homomorphism Z(C) → Z(EndA A) = Z(Aopp ) = Z(A), where Z(A) is the center of A.

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An Overlooked Infinite System of Groups of Order pq2 by Miller G. A.


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