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Linkage principle for small quantum groups

Cristian Vay

TL;DR

This work establishes a linkage principle for small quantum groups realized as Drinfeld doubles of Nichols algebras of diagonal type. By adapting Andersen–Jantzen–Soergel techniques and employing Lusztig isomorphisms across Weyl groupoids, it constructs twisted Verma modules, analyzes their morphisms, and derives a block decomposition together with a notion of typicality akin to Lie superalgebras. The main results show that composition factors of Verma modules lie in orbits under a generalized dot-action, classify blocks, and provide a character formula for 1-atypical simples, with typical simples being simple and projective Verma modules. The work connects small quantum groups to modular and super-type representation theory, and extends the AJS categorical framework to this broader setting, enabling potential applications to Braided Drinfeld doubles and related quantum algebras. Overall, it deepens understanding of representation theory for finite-dimensional Hopf algebras arising from diagonal Nichols algebras and clarifies how classical and superalgebra phenomena manifest in this quantum context.

Abstract

We consider small quantum groups with root systems of Cartan, super and modular type, among others. These are constructed as Drinfeld doubles of finite-dimensional Nichols algebras of diagonal type. We prove a linkage principle for them by adapting techniques from the work of Andersen, Jantzen and Soergel in the context of small quantum groups at roots of unity. Consequently we characterize the blocks of the category of modules. We also find a notion of (a)typicality similar to the one in the representation theory of Lie superalgebras. The typical simple modules turn out to be the simple and projective Verma modules. Moreover, we deduce a character formula for 1-atypical simple modules.

Linkage principle for small quantum groups

TL;DR

This work establishes a linkage principle for small quantum groups realized as Drinfeld doubles of Nichols algebras of diagonal type. By adapting Andersen–Jantzen–Soergel techniques and employing Lusztig isomorphisms across Weyl groupoids, it constructs twisted Verma modules, analyzes their morphisms, and derives a block decomposition together with a notion of typicality akin to Lie superalgebras. The main results show that composition factors of Verma modules lie in orbits under a generalized dot-action, classify blocks, and provide a character formula for 1-atypical simples, with typical simples being simple and projective Verma modules. The work connects small quantum groups to modular and super-type representation theory, and extends the AJS categorical framework to this broader setting, enabling potential applications to Braided Drinfeld doubles and related quantum algebras. Overall, it deepens understanding of representation theory for finite-dimensional Hopf algebras arising from diagonal Nichols algebras and clarifies how classical and superalgebra phenomena manifest in this quantum context.

Abstract

We consider small quantum groups with root systems of Cartan, super and modular type, among others. These are constructed as Drinfeld doubles of finite-dimensional Nichols algebras of diagonal type. We prove a linkage principle for them by adapting techniques from the work of Andersen, Jantzen and Soergel in the context of small quantum groups at roots of unity. Consequently we characterize the blocks of the category of modules. We also find a notion of (a)typicality similar to the one in the representation theory of Lie superalgebras. The typical simple modules turn out to be the simple and projective Verma modules. Moreover, we deduce a character formula for 1-atypical simple modules.
Paper Structure (40 sections, 42 theorems, 185 equations, 4 figures)

This paper contains 40 sections, 42 theorems, 185 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Sketch of the proof
  4. Relations with other algebras in the literature
  5. Organization
  6. Acknowledgments
  7. Conventions
  8. Finite-dimensional Nichols algebras of diagonal type
  9. PBW basis
  10. Weyl groupoid
  11. Root systems
  12. A shift operation on weights
  13. Small quantum groups
  14. Lusztig isomorphisms
  15. Parabolic subalgebras
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $L$ be a composition factor of $M(\pi\widetilde{\mu})$, $\mu\in\mathbb Z^\mathbb{I}$. Then $L\simeq L(\pi\widetilde{\mu})$ or $L\simeq L(\pi\widetilde{\lambda})$ with $\lambda=\beta_r\downarrow\cdots\beta_1\downarrow\mu$ for some $\beta_1, ..., \beta_r\in\Delta_+^\mathfrak{q}$.

Figures (4)

  • Figure 1: Algebras involved in Lusztig's conjectures. Given a finite root system $\Delta$, we write $\mathfrak{g}$ and $\mathfrak{g}_k$ for the associated Lie algebras over $\mathbb{C}$ and over an algebraically closed field $k$ of characteristic $p$ odd ($\neq3$ if $\mathfrak{g}$ has a component of type $G_2$). We write $q$ for a $p$-th root of unity in $\mathbb{C}$.
  • Figure 2: We construct small quantum groups from matrices with finite-dimensional Nichols algebras of diagonal type; $\Gamma$ is a group quotient of $\mathbb Z^\theta$. For instance, the positive part of $u_q(\mathfrak{g})$ is the Nichols algebra of $\mathfrak{q}=(q^{d_ic_{ij}})_{i,j}$ where $C=(c_{ij})_{i,j}$ is the Cartan matrix of $\mathfrak{g}$ and $(d_ic_{ij})_{i,j}$ is symmetric. Thus, the corresponding bosonization is the Borel subalgebra and $u_q(\mathfrak{g})$ is a quotient of $u_{\mathfrak{q}}$ by a central group subalgebra.
  • Figure 3: Here $w_0=1^{\mathfrak{q}}\sigma_{i_1}\cdots\sigma_{i_n}$ is a reduced expression of the longest element in ${}^\mathfrak{q}\mathcal{W}$, $w=1^\mathfrak{q}\sigma_{i_1}\cdots\sigma_{i_{s-1}}$ and $\beta=w\alpha_{i_s}\in\Delta_+^\mathfrak{q}$. The horizontal maps are generators of the corresponding Hom-spaces and ${\mathrm{Im}} \psi=\mathop{\mathrm{ {\mathrm{Ker}} }}\nolimits\varphi$.
  • Figure 4: Let $w_0=1^{\mathfrak{q}}\sigma_{i_1}\cdots\sigma_{i_n}$ be a reduced expression of the longest element in ${}^\mathfrak{q}\mathcal{W}$. We set $w_s=1^\mathfrak{q}\sigma_{i_1}\cdots\sigma_{i_{s-1}}$ and $\beta_s=w_s\alpha_{i_s}$, $1\leq s\leq n+1$. The morphisms $\varphi_i$ are constructed as in §\ref{['sss:a generator']} with $x=1^\mathfrak{q}$ and $w=w_0$. The morphism $\psi$ is given by Lemma \ref{['le:psi w']} with $t_s=t_{\beta_s}^\pi(\mu\langle w_s\rangle)$. Moreover, $t_s=n_{\beta_s}^\pi(\mu)$ by Lemma \ref{['le:n beta = t beta']}. Then the compositions $\varphi_s\cdots\varphi_1$ are generators of the corresponding Hom spaces by Corollary \ref{['cor:a generator']} and $\Phi=\varphi_n\cdots\varphi_1$ satisfies Lemma \ref{['le:imagen de Z w0 mu w0 en Z mu']}. The morphisms $\tilde{\varphi}_i$ are constructed analogously starting from $\mu-t_s\beta_s$ instead of $\mu$.

Theorems & Definitions (97)

  • Theorem 1.1: Strong linkage principle
  • Corollary 1.2: Linkage principle
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Definition 3.6
  • Example 3.7
  • Definition 4.1
  • ...and 87 more