A non-semisimple Witt class
We describe several infinite families of braided finite tensor categories. A simplest example gives a non-degenerate braided tensor category which is not Witt equivalent to a semisimple category.
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arXiv subjects
Alexandra Utiralova
Publications and source records attributed to Alexandra Utiralova.
The highest weight theory for Representations of General Linear groups in the Verlinde categories in positive characteristic
Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups $GL(X)$ in the Verlinde category $Ver_p$ in characteristic $p$. The main question we answer is how to translate between highest weight labelings for different choices of the Borel subgroup $B(X)\subset GL(X)$. We do this by reducing the general case to the study of representations of the group $GL(X)$ for $X=L_m\oplus L_{n}$ using the method of odd reflections. On the category of representations of $GL(L_m\oplus L_{n})$ we introduce the structure of the highest weight category, as well as the categorical action of $\widehat{\mathfrak{sl}}_p$ through translation functors. It allows us to understand projective and injective objects, BGG reciprocity, duality and lowest weights for simple modules, and standard filtration multiplicities for projective objects.
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Harish-Chandra bimodules of finite $K$-type in Deligne categories
We continue the study of Harish-Chandra bimodules in the setting of the Deligne categories $\mathrm{Rep}(G_t)$, that was started in the previous work of the first author (arXiv:2002.01555). In this work we construct a family of Harish-Chandra bimodules that generalize simple finite dimensional bimodules in the classical case. It turns out that they have finite $K$-type, which is a non-vacuous condition for the Harish-Chandra bimodules in $\mathrm{Rep}(G_t)$. The full classification of (simple) finite $K$-type bimodules is yet unknown. This construction also yields some examples of central characters $χ$ of the universal enveloping algebra $U(\mathfrak{g}_t)$ for which the quotient $U_χ$ is not simple, and, thereby, it allows us to partially solve a question posed by Pavel Etingof in one of his works.
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Harish-Chandra bimodules in the Deligne category $\text{Rep}(GL_t)$
In this paper we study the category of Harish-Chandra bimodules $HC_{χ,ψ}$ in the Deligne category $\text{Rep}(GL_t)$. In particular, we answer Question 3.25 posed in Pavel Etingof's paper arXiv:1407.0373 and determine for which central characters $χ$ and $ψ$ this category is not zero.
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Olshanski's Centralizer Construction and Deligne Tensor Categories
The family of Deligne tensor categories $\mathrm{Rep}(GL_t)$ is obtained from the categories $\mathbf{Rep}~GL(n)$ of finite dimensional representations of groups $GL(n)$ by interpolating the integer parameter $n$ to complex values. Therefore, it is a valuable tool for generalizing classical statements of representation theory. In this work we introduce and prove the generalization of Olshanski's centralizer construction of the Yangian $Y(\mathfrak{gl}_n)$. Namely, we prove that for generic $t\in\mathbb{C}$ the centralizer subalgebra of $GL_t$-invariants in the universal enveloping algebra $U(\mathfrak{gl_{t+n}})$ is the tensor product of $Y(\mathfrak{gl}_n)$ and the center of $U(\mathfrak{gl_{t}})$. The main feature of this construction is that it does not involve passing to a limit, contrary to the original construction of Olshanski.
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