2016/11/11 by Andrew Prudhom, Prudhom, Andrew, Alexander Varchenko +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1611.03944
We consider the Frobenius algebra of functions on the critical set of the\nmaster function of a weighted arrangement of hyperplanes in Ck with normal\ncrossings. We construct two potential functions (of first and second kind) of\nvariables labeled by hyperplanes of the arrangement and prove that the matrix\ncoefficients of the Grothendieck residue bilinear form on the algebra are given\nby the 2k-th derivatives of the potential function of first kind and the\nmatrix coefficients of the multiplication operators on the algebra are given by\nthe (2k+1)-st derivatives of the potential function of second kind. Thus the\ntwo potentials completely determine the Frobenius algebra. The presence of\nthese potentials is a manifestation of a Frobenius like structure similar to\nthe Frobenius manifold structure.\n We introduce the notion of an elementary subarrangement of an arrangement\nwith normal crossings. It turns out that our potential functions are local in\nthe sense that the potential functions are sums of contributions from\nelementary subarrangements of the given arrangement. This is a new phenomenon\nof locality of the Grothendieck residue bilinear form and multiplication on the\nalgebra.\n It is known that this Frobenius algebra of functions on the critical set is\nisomorphic to the Bethe algebra of this arrangement. (That Bethe algebra is an\nanalog of the Bethe algebras in the theory of quantum integrable models.) Thus\nour potential functions describe that Bethe algebra too.\n