2018/12/02 by Ben Cox, Cox, Ben, Xiangqian Guo +5
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math-ph #math.MP #math.RT
paper · pdf · doi:10.48550/arxiv.1812.00330
24 pages, submitted
arxiv created 2018/12/02 · openalex publication_date 2018/12/02 · arxiv updated 2018/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakg⊗ R be the corresponding current Lie algebra. \colorblack Here \mathfrak g is a finite dimensional simple Lie algebra defined over \mathbb C and p(t)= t(t-α1)⋯ (t-α2n)=∑k=12n+1aktk. In earlier work, Cox and Im gave a generator and relations description of the universal central extension of \mathfrakg⊗ 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. We give examples of 2n-tuples (α1,…,α2n), which are the automorphism groups \mathbb Gn=Dicn, \mathbb Un≅ Dn (n odd), or \mathbb Un (n even) of the hyperelliptic curves S=ℂ[t, u: u2 = t(t-α1)⋯ (t-α2n)] given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is \mathbb Un=Dn, where n is odd.