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Algebra Seminar


Tid: 2022-01-26 16:00 till 17:00
Plats: The talk will be given on Zoom. To receive the link, please contact the organiser.
Kontakt: anitha [dot] thillaisundaram [at] math [dot] lu [dot] se
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Speaker: Marina Avitabile (University of Milan-Bicocca)


Title: Generalized finite polylogarithms


Abstract: I will introduce a generalization L_1^{\alpha}(X) of the truncated logarithm L_1(X) = \sum_{k = 1}^{p-1} X^k/k in prime characteristic p, which depends on a parameter \alpha. The main motivation of this study is L_1^{\alpha}(X) being an inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential given by certain Laguerre polynomials. Such Laguerre polynomials play a role in a grading switching technique for non-associative algebras - whose aim is to produce a new grading of an algebra from a given one - because they satisfy a weak analogue of the functional equation exp(X) exp(Y) = exp(X+Y ) of the exponential series.

I will also present some functional equations satisfied by L_1^{\alpha}(X) motivated by known functional equations for L_1(X) = L_1^{0}(X). Finally I will discuss a generalization L_d^{\alpha}(X) of the finite polylogarithms L_d^{0}(X) = L_d(X) = \sum_{k = 1}^{p-1} X^k/k^d, where d is an integer.

All the results mentioned are a joint work with Sandro Mattarei.