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Differential Geometry Seminar Abstract

Ted Voronov (Manchester)

The Berezinian of matrices, exterior powers and recurrence relations

The Berezinian is the analog of the determinant for the super case. Unlike determinant, it is a rational function of the matrix (not a polynomial). At the first glance, there is no connection of it with the exterior powers. In the supercase there are infinitely many exterior powers and no `top' power, in contrast with the familiar purely even situation. However, as we show, this connection exists and, loosely, reminds an analytic continuation from a neighborhood of zero to infinity.

We obtained universal recurrence relations that hold for supertraces of exterior powers of a linear operator; they are underlied by relations in Grothendieck ring. They come from expansions of the characteristic (rational) function Ber (1+z A) and lead to a new beautiful formula expressing the Berezinian as a ratio of Hankel determinants made of supertraces.

See: arxiv.org/abs/math.DG/0309188

(Joint work with Hovhannes Khudaverdian.)