[Paper Review] Cuspidal $\mathfrak{sl}_n$-modules and deformations of certain Brauer tree algebras

Volodymyr Mazorchuk, Catharina Stroppel|arXiv (Cornell University)|Jan 15, 2010

Algebraic structures and combinatorial models34 references4 citations

TL;DR

This paper establishes that blocks of cuspidal weight and generalized weight modules for $χ_n$ are isomorphic to finite-dimensional modules over one- or multi-parameter deformations of specific Brauer tree algebras, which are shown to be nontrivial deformations of the path algebra $A^{n-1}$ via Hochschild cohomology. The key contribution is the explicit realization of these deformations as universal one-parameter (or multi-parameter) deformations of self-injective symmetric algebras arising in representation theory and algebraic geometry.

ABSTRACT

We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are one-parameter (respectively multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.

Motivation & Objective

• To classify blocks of the category of cuspidal (generalized) weight modules for $χ_n$ using associative algebras.
• To show that these algebras are deformations of Brauer tree algebras, specifically $A^{n-1}$, arising from centralizer subalgebras in parabolic category $ϵ$.
• To establish that the deformations are nontrivial by analyzing their infinitesimal structure via Hochschild cohomology.
• To unify diverse appearances of $A^{n-1}$ in representation theory—such as in Temperley-Lieb algebras, Khovanov algebras, and Milnor fibers—by identifying their deformation theory.

Proposed method

• The authors construct a functor $\mathrm{F}$ from $\mathbb{C}[[x_1,\dots,x_n]]$-modules to the category of cuspidal modules $\hat{\mathcal{C}}_{\lambda,\xi}$, establishing a link between power series rings and module categories.
• They use Gelfand-Zelevinsky and Gelfand-Zelevinsky realizations to analyze the structure of cuspidal modules and their blocks.
• The deformation theory of $A^{n-1}$ is analyzed via Hochschild cohomology, particularly $H^2(A^{n-1}, A^{n-1})$, to classify one- and multi-parameter deformations.
• Nontriviality of the deformations is proven by showing that the image of $x = \beta_1\alpha_1$ in $D_{\lambda,\xi}$ satisfies $x^2 \neq 0$ modulo $\mathfrak{m}^2$, implying nonvanishing second cohomology.
• The authors use the restricted dual and self-adjoint functors to construct projective-injective modules and analyze their Loewy filtrations to verify nonvanishing compositions.
• The universal property of deformations is applied via Corollary 30, showing that any nontrivial deformation over $\mathbb{C}[[x]]$ or $\mathbb{C}[[x_1,\dots,x_n]]$ must be isomorphic to the algebra described in Theorem 1.

Experimental results

Research questions

• RQ1 How are blocks of cuspidal weight modules for $\mathfrak{sl}_n$ classified in terms of associative algebras?
• RQ2 What is the nature of the deformation of the Brauer tree algebra $A^{n-1}$ that arises in the regular block of cuspidal weight modules?
• RQ3 Are these deformations nontrivial, and how can their nontriviality be detected via cohomological invariants?
• RQ4 How do these algebras relate to other known algebras such as those in Temperley-Lieb algebras, Khovanov algebras, or Schur algebras?
• RQ5 Can the deformation of $A^{n-1}$ be realized as a universal one- or multi-parameter deformation in the context of category $\mathcal{C}$?

Key findings

• Every non-integral or singular block of the category $\mathcal{C}$ of cuspidal weight $\mathfrak{sl}_n$-modules is equivalent to the category of finite-dimensional modules over $\mathbb{C}[[x]]$, a formal power series ring.
• Every non-integral or singular block of the generalized weight category $\hat{\mathcal{C}}$ is equivalent to the category of finite-dimensional modules over $\mathbb{C}[[x_1,\dots,x_n]]$, a multivariate power series ring.
• For $n > 2$, every integral regular block of $\mathcal{C}$ is equivalent to the category of finite-dimensional modules over a flat one-parameter deformation of $A^{n-1}$, which is nontrivial as an infinitesimal deformation.
• The deformation of $A^{n-1}$ is nontrivial when reduced modulo $\mathfrak{m}^2$, as shown by the nonvanishing of $x^2$ in $D_{\lambda,\xi}$ for $x = \beta_1\alpha_1$, implying nontrivial Hochschild cohomology.
• The algebra $D_{\lambda,\xi}$ is a universal one-parameter deformation of $A^{n-1}$ over $\mathbb{C}[[x]]$, and the same holds for the multi-parameter case over $\mathbb{C}[[x_1,\dots,x_n]]$.
• The results unify various appearances of $A^{n-1}$ in representation theory: as centralizer subalgebras in parabolic category $\mathcal{O}$, in generalized Khovanov algebras, and in the Milnor fiber of Kleinian singularities.