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Berezinians of supermatrices, exterior products of superspaces and recurrent sequences

Le : 12/10/2015 14h15
Par : Hovhannes Khudaverdian (Manchester)
Lieu : I 001
Lien web :
Résumé : In supermathematics, commuting (even) and anticommuting (odd) variables are considered on an equal footing. Integration rules for anticommuting odd variables discovered by F.A. Berezin imply that an analogue of the Jacobian, the superdeterminant (Berezinian) of a transformation of even and odd variables, is not a polynomial in the partial derivatives, but a rational function. It is well known that, for a usual n-dimensional vector space V, the determinant of a linear operator A is related with the traces of exterior powers of A via Cayley-Hamilton type identities involving the characteristic polynomial R_A(z)=det(1+Az) and its coefficients. In the supercase, the characteristic function R_A(z)=Ber(1+Az) is no longer a polynomial. We study power expansions of the characteristic function R_A(z) for a linear operator A acting on p|q-dimensional superspace V. We show that the traces of exterior powers of A satisfy recurrence relations of period q. (Eg. for q=1 they form a geometric progression and for q=2 a Fibonacci sequence.) This leads us to the explicit formula expressing the Berezinian of a linear operator as a rational function of traces. We also study the geometrical meaning of the Kramer rule and explain how it works in the supercase. (Joint work with Th.Th. Voronov.)