Abstract
We give a description of the bimodule of double derivations of a finite dimensional semi-simple algebra S and its double Schouten bracket in terms of a quiver. This description is used to determine which degree two monomials induce double Poisson brackets on S. In case S = ℂ⊕n, a criterion for any degree two element to give a double Poisson bracket is deduced. For S = ℂ⊕n and S′ = ℂ⊕m the induced Poisson bracket on the variety of isomorphism classes of semi-simple representations iss n (S * T) of the free product S * T is given.
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The author is postdoctoral fellow of the Fund for Scientific Research—Flanders (F.W.O.-Vlaanderen)(Belgium). Part of the work presented here was done while the author was visiting the University of Leeds, supported by a travel grant from the Fund for Scientific Research—Flanders (F.W.O.-Vlaanderen)(Belgium). The author would like to thank his hosts for their hospitality.
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Weyer, G.V.d. Double Poisson Structures on Finite Dimensional Semi-Simple Algebras. Algebr Represent Theor 11, 437–460 (2008). https://doi.org/10.1007/s10468-008-9088-3
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DOI: https://doi.org/10.1007/s10468-008-9088-3