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January 18 Math Colloquium

Ben Cox, CofC

On The Module Structure Of The Center Of Hyperelliptic Krichever-Novikov Algebras

Let R := R2(p) = C[t±1, u | u2 = t(t − α1)· · ·(t − α2n)] be the coordinate ring of a nonsingular hyperelliptic curve and let g ⊗ R be the corresponding current Lie algebra where α1, . . . , α2n are distinct complex numbers. Here g is a finite dimensional simple Lie algebra (such as the vector space of 2 × 2 matrices defined over C of trace zero). In earlier work the first two authors gave a generator and relations description of the universal central extension of g ⊗ R in terms of certain families of polynomials Pk,i and Qk,i and they described how the center ΩR/dR of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group C2k or the dihedral group D2k. In the talk we will describe this decomposition when the automorphism group is Gn = Dicn, and Un = Dn (n odd) or Un (n even). We also give examples of 2n-tuples (α1, . . . , α2n) that give rise to these groups. More precisely V2n, Gn = Dicn, and Un∼= Dn (n odd) or Un are the automorphism groups of the hyperelliptic curves 

S = C[t, u | u2 = t(t − α1)· · ·(t − α2n)] 

given in earlier work of Cox, Guo and Zhao.

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