We prove an analogue of Kac’s Theorem, describing the dimension types of indecomposable coherent sheaves (or parabolic bundles) over weighted projective lines in terms of root systems for loop algebras of Kac–Moody Lie algebras. We use a theorem of Peng and Xiao to associate a Lie algebra to the category of coherent sheaves for a weighted projective line over a finite field, and find elements of this Lie algebra which satisfy the relations defining the loop algebra. We use these elements in the proof of our analogue of Kac’s Theorem.
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William Crawley-Boevey, Kac's Theorem for weighted projective lines. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1331–1345DOI 10.4171/JEMS/232