Annular webs and Levi subalgebras

TL;DR

The paper constructs an annular web category quotient that models the representation theory of Levi subalgebras of gl_n, providing a new annular skew Howe duality framework and a web version of the generalized blob algebra.

Contribution

It introduces a novel quotient of annular quantum gl_n webs that captures the representation category of Levi subalgebras, extending skew Howe duality to an annular setting.

Findings

Establishes an equivalence between the quotient web category and representations of Levi subalgebras.

Provides a web-based description of the generalized blob algebra.

Offers a new perspective on categorifying Levi subalgebra representations.

Abstract

For any Levi subalgebra of the form $\mathfrak{l}=\mathfrak{gl}_{l_{1}}\oplus\dots\oplus\mathfrak{gl}_{l_{d}}\subseteq\mathfrak{gl}_{n}$ we construct a quotient of the category of annular quantum $\mathfrak{gl}_{n}$ webs that is equivalent to the category of finite dimensional representations of quantum $\mathfrak{l}$ generated by exterior powers of the vector representation. This can be interpreted as an annular version of skew Howe duality, gives a description of the representation category of $\mathfrak{l}$ by additive idempotent completion, and a web version of the generalized blob algebra.