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Seminários do IMPA

Física Matemática

Invariant subalgebras in a skew-group ring and their modules
Elizaveta Vishnyakova

Quinta-feira, 9 de maio de 2019, 15:30
Sala 236

Gelfand and Zeitlin constructed a basis in finite dimensional $\mathfrak{gl}_n(\mathbb C)$-modules together with  explicit formulas for  $\mathfrak{gl}_n(\mathbb C)$-action. These formulas for $\mathfrak{gl}_n(\mathbb C)$-action are called classical Gelfand—Zeitlin formulas. Later it was noticed by Gelfand and Graev that the classical Gelfand—Zeitlin formulas may be used to obtain a family of infinite dimensional   $\mathfrak{gl}_n(\mathbb C)$-modules. More general Gelfand—Zetlin modules are developed by Drozd, Ovsienko and Futorny.

The main difficulty here was to construct and classify singular Gelfand—Zeitlin modules. That is, Gelfand—Zeitlin modules where the (rational) coefficients of the classical Gelfand—Zeitlin formulas have potential singularities. A significant step in this direction was done in 2017—2018 by Ramirez, Zadunaisky and by Early, Mazorchuk, E.V.. For instance Early, Mazorchuk and E.V. constructed all simple Gelfand—Zeitlin $\mathfrak {gl}_n(\mathbb C)$-modules. A classification of  simple Gelfand—Zeitlin $\mathfrak {gl}_n(\mathbb C)$-modules was obtained by Ben Webster in 2019.

We will discuss our most recent paper with Mazorchuk that is devoted to a generalization of Gelfand—Zeitlin theory to any invariant subalgebras in a skew-group ring.