International Conference "Dualities in Quantum Groups"

Introduction to total positivity

An invertible $n \times n$ real matrix is called totally positive if all its minors are positive. The study of such matrices dates back to Schoenberg and Grantmacher-Krein in 1930s. The theory has been generalized by Lusztig in 1994 to arbitrary split reductive connected groups (from general linear groups). This generalization depends on deep results from the theory of canonical bases arising from quantum groups. Since then, the theory of total positivity has found numerous applications, including cluster algebras, higher Teichmuller theory, the physics of scattering amplitudes, etc. The goal of this course is give an introduction to the theory of total positivity.

Super Duality for finite W-algebras and W-superalgebras

We describe an analogue of super duality for Lie algebras and Lie superalgebras in the setting of finite W-algebras and W-superalgebras in some suitable W-(super)algebra module categories analogous to parabolic BGG categories for Lie (super)algebras introduced originally by Brundan-Goodwin-Kleshchev. This, in a sense, is a generalization of super duality for Lie (super)algebras. This is a work in progress jointly with Weiqiang Wang.

Classification of irreducible representations of real quasi-reductive algebraic supergroups under some working hypotheses

Loewy gave a classification scheme of real finite dimensional irreducible representations of groups $G$ in terms of complex finite dimensional irreducible representations of $G$ and their complex conjugation. We can determine the classification of irreducible representations of connected real reductive algebraic groups numerically by application of Loewy's scheme and the Borel–Weil theorem. The key is to put the complex conjugation $\bar{B}'$ of a chosen complex Borel subgroup $B'$ back to $B'$ through a Weyl group twist. In this talk, I explain Loewy's scheme. Then we see that a similar classification result holds for real quasi-reductive algebraic supergroups under reasonable working hypotheses by Shibata's Borel–Weil theory. I will show examples of real forms of $\mathrm{GL}_{m|n}$, $\mathrm{SpO}_{m|n}$, and $P_n$ which do and do not satisfy the hypotheses. If time permits, we see briefly how to deal with the classification problem of irreducible representations of real quasi-reductive algebraic supergroups without the working hypotheses.

Introduction to super duality

The theory of super duality is about an equivalence between two parabolic BGG categories for Lie algebras and Lie superalgebras. As an application, it provides a solution to the irreducible characters of Lie superalgebras of type ABCD in terms of the Kazhdan-Lusztig formula for Lie algebras. This lecture briefly reviews the theory of super duality and related works.

Spin-oscillator correspondence for quantum affine algebras and super duality

We establish a quantum affine analogue of a correspondence between irreducible finite-dimensional and irreducible oscillator representations of Lie algebras of type C/D and D/C (and also for type B) arising from Howe's reductive dual pairs. This can be further explained by introducing an interpolating module category of quantum affine superalgebras, which should be viewed as a manifestation of super duality, while our results crucially make use of monoidal structures and R-matrices. This talk is based on an ongoing project with Jae-Hoon Kwon and Masato Okado.

Drinfeld type presentation of quasi-split affine i-quantum groups

A quantum symmetric pair consists of a quantum group and its coideal subalgebra (called an i-quantum group). A quantum group can be viewed as an example of i-quantum groups associated to symmetric pairs of diagonal type. In this talk, I shall give a Drinfeld type presentation for quasi-split affine i-quantum groups, and use Hall algebras of (weighted) projective lines to realize this presentation. This is based on joint works with Shiquan Ruan, Weiqiang Wang, Weinan Zhang.

Howe duality and invariant theory of i-quantum groups of type AIII

It is known that for classical groups, the Schur duality, the Howe duality and the fundamental theorems of invariant theory are equivalent, though they are not equally straightforward. The same picture was not completed for quantum analogues except the general linear quantum groups, whose Schur duality, Howe duality and fundamental theorems of invariant theory was established by Jimbo, Zhang and Lehrer-Zhang-Zhang, respectively. In this talk, we shall provide the Howe duality and the fundamental theorems of invariant theory for the i-quantum groups of type AIII, which can be regarded as the case of quantum type B/C in the sense of Hecke algebras. This is joint work with Zheming Xu.

Quantum supersymmetric pairs and $\imath$Schur duality of type AI-II

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\theta$ be an involution of $\mathfrak{g}$. The quantization $(U_q(\mathfrak{g}), U^\imath)$ of the symmetric pair $(\mathfrak{g}, \mathfrak{g}^\theta)$ was systematically developed by Letzter where $U_q(\mathfrak{g})$ is the Drinfeld-Jimbo quantum group and $U^\imath$ is a coideal subalgebra of it. We ususally refer $U^\imath$ as the $\imath$quantum group. Over the last decade, many fundamental constructions in quantum groups have been generalized to $\imath$quantum groups by Wang and his collaborators. In this talk, we will discuss the super analogue of $U^\imath$ and introduce one specific family which unites $\imath$quantum groups (non-super) of type AI and AII. We will also demonstrate an $\imath$Schur duality between this specific family and the $q$-Brauer algebra. This duality can be viewed as a quantization of the classical duality between the orthosymplectic Lie superalgebra and the Brauer algebra. This is joint work with Weiqiang Wang.

On representations of split quasireductive supergroups

Representation theory of algebraic supergroups over a field of positive characteristic has applications to non-super modular representations, making it more than just a generalization and quite interesting. However, compared to the non-super case, the behavior of super root systems is quite special, and that's why the representation theory does not always proceed in parallel. In this talk, first, we will review a framework of characteristic-free study of structures and representations of algebraic supergroups via a Hopf-algebraic approach. Then we will study the behavior of the sheaf cohomology of a split quasireductive supergroup and construct all irreducible representations.

Canonical bases and super Kazhdan-Lusztig ABC

It is well known that the irreducible character formulas for the BGG category of semisimple Lie algebras are formulated in terms of Kazhdan-Lusztig bases of Hecke algebras. For the irreducible character formulas in BGG category for Lie superalgebras of type A (and respectively, BCD), KL bases for Hecke algebras are insufficient and must be replaced by canonical bases on modules for quantum groups of type A (and respectively, i-quantum groups). The transition from Hecke algebras to quantum groups or i-quantum groups is made possible by Schur-Jimbo duality or i-Schur duality. This has further stimulated the development of a general theory of canonical bases arising from quantum symmetric pairs. In this lecture series, I will explain some of these constructions, based on work of Brundan and joint work with Cheng and Lam (for super A) and with Bao (on super BCD).

Typical irreducible character formula for generalized quantum groups

(1) I explain the typical irreducible character formula for generalized quantum groups (J. Algebra Appl. 20 (2021), no. 1, 2140014). There are two finite dimensional irreducible highest weight modules $V(\lambda)$ and $V(\mu)$ of some generalized quantum groups such that $V(\lambda)$ and $V(\mu)$ are typical and non-typical respectively, and their highest weights $\lambda$ and $\mu$ look very similar, but their characters are really different (Appendix of arXiv:1909.08881). (2) I also explain Hamiltonian cycles of the Cayley graphs of the finite Weyl groupoids (arXiv:2310.12543 (also Toyama. Math. J. 43 (2022), 1-76)). This is a joint work with Takato Inoue.