Seminar

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We consider quadratic deformations of the q-symmetric algebras A_q given by x_i x_j = q_{ij} x_j x_i, for a matrix of q's satisfying q_{ij} q_{ji} = 1, in variables x_1, ..., x_n with n odd.  We explicitly describe the Hochschild cohomology and compute the weights of the torus action (dilating the x_i variables). Under a mild condition on q, we describe quadratic deformations which restrict to the universal formal deformations. These are Koszul and Calabi—Yau (hence Artin—Schelter regular).  The deformations are indexed by "smoothing diagrams", a collection of disjoint cycles and segments in the complete graph on n vertices, viewed as the dual complex for the coordinate hyperplanes in P^{n-1}.  Already for n=5 there are 40 of these, most of which define apparently new families of quadratic algebras.  The algebras are obtained by a tensor product of a quadratic filtered deformation for the segment part, and Feigin—Odesskii elliptic algebras for the cycle parts. Our proof also applies to toric log symplectic structures on P^{n-1} (the quasiclassical analogue), recovering a special case of our previous results for general log symplectic varieties with normal crossings divisors, which motivated this project.  This is joint work with Mykola Matviichuk and Brent Pym.