[Paper Review] Irreducible Integrable Representations of Toroidal Lie Algebras

Tanusree Khandai|arXiv (Cornell University)|Jul 15, 2016

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper provides an alternative proof for the classification of irreducible integrable representations of toroidal Lie algebras with finite-dimensional weight spaces, using the integral form of the universal enveloping algebra of affine Kac-Moody algebras and current algebra actions. It establishes a parametrization of isomorphism classes via orbits of $({ m C}^*)^{k-1}$ acting on finitely supported functions to dominant integrable weights of affine Lie algebras.

ABSTRACT

The irreducible integrable representations with finite-dimensional weight spaces of toroidal Lie algebras on which the center acts non-trivially were classified by S.Eswara Rao. In this paper we give a compact proof of the results that lead to the classification theorem. Further we establish a necessary and sufficient condition under which two such irreducible modules are isomorphic.

Motivation & Objective

• To provide an alternative proof for the classification of irreducible integrable modules with finite-dimensional weight spaces in the category $\mathcal{I}_{\text{fin}}^{\ast}$ for toroidal Lie algebras.
• To parametrize the isomorphism classes of such modules using the action of $({\mathbf{C}}^\ast)^{k-1}$ on the set of finitely supported functions to dominant integrable weights of the associated affine Kac-Moody algebra.
• To clarify the role of the infinite-dimensional $\mathbf{Z}^k$-graded center in the representation theory of toroidal Lie algebras, particularly in restricting non-trivial central actions.
• To streamline the homological study of objects in $\mathcal{I}_{\text{fin}}^{\ast}$ using methods aligned with existing frameworks in affine Lie algebra representation theory.
• To establish a necessary and sufficient condition for isomorphism between two irreducible modules in $\mathcal{I}_{\text{fin}}^{\ast}$ via group action and module structure.

Proposed method

• Utilizes the integral form of the universal enveloping algebra $\mathcal{U}(\mathfrak{g}_{\text{aff}})$ of the affine Kac-Moody algebra $\mathfrak{g}_{\text{aff}}$ to analyze module structures.
• Applies the action of current algebras $\mathfrak{g}_{\text{fin}} \otimes \mathbf{C}[t_1^{\pm 1}]$ on weight vectors to derive constraints on central actions.
• Employs isomorphisms between modules induced by automorphisms $s_{\mathbf{b}}$ of the polynomial algebra $\mathbf{C}[t_2^{\pm 1}, \dots, t_k^{\pm 1}]$ corresponding to $\mathbf{b} \in (\mathbf{C}^\ast)^{k-1}$.
• Analyzes the support of weight functions $\pi$ and their images under $s_{\mathbf{b}}$, linking module isomorphism to orbit equivalence under the group action.
• Relies on the classification of irreducible integrable $\mathfrak{g}_{\text{aff}}$-modules with finite-dimensional weight spaces from [C1, CFK, CP3] as foundational input.
• Uses the fact that each graded component of the center $\mathcal{Z}$ of the toroidal algebra admits at most one non-trivial action in $\mathcal{I}_{\text{fin}}^{\ast}$, proven via current algebra actions and module decomposition.

Experimental results

Research questions

• RQ1 What is the complete parametrization of isomorphism classes of irreducible integrable $\mathcal{T}(\mathfrak{g})$-modules with finite-dimensional weight spaces?
• RQ2 How does the action of the $\mathbf{Z}^k$-graded center constrain the structure of such modules, and why can at most one generator per graded component act non-trivially?
• RQ3 Under what conditions are two such modules isomorphic, and how can this be characterized algebraically?
• RQ4 Can the classification of these modules be re-derived using the integral form of $\mathcal{U}(\mathfrak{g}_{\text{aff}})$ and current algebra actions, rather than Heisenberg algebra results?
• RQ5 How does the $({\mathbf{C}}^\ast)^{k-1}$-action on the set of finitely supported functions to $P_{\text{aff}}^+$ classify the isomorphism classes of these modules?

Key findings

• Every irreducible $\mathcal{T}(\mathfrak{g})$-module in $\mathcal{I}_{\text{fin}}^{\ast}$ is isomorphic to a twist of a module $X_{\pi}^{\mathbf{g}}$ for some $\pi \in \Pi$ and $\mathbf{g} \in \mathbf{Z}^{k-1}$, up to one-dimensional module twists.
• The isomorphism class of $X_{\pi}^{\mathbf{g}}$ is determined by the orbit of $\pi$ under the action of $({\mathbf{C}}^\ast)^{k-1}$ on $\Pi$, with $\mathbf{g}$ well-defined modulo $G_\pi$.
• Two modules $X_{\pi}^{\mathbf{g}}$ and $X_{\pi'}^{\mathbf{g}'}$ are isomorphic if and only if there exists $\mathbf{b} \in ({\mathbf{C}}^\ast)^{k-1}$ such that $\operatorname{supp}(\pi') = \{s_{\mathbf{b}}(M) \mid M \in \operatorname{supp}(\pi)\}$ and $X(\pi(M)) \cong X(\pi'(s_{\mathbf{b}}(M)))$ as $\mathfrak{g}_{\text{fin}} \otimes \mathbf{C}[t_1^{\pm 1}] \oplus \mathbf{C}K_1$-modules.
• The center $\mathcal{Z}$ of $\mathcal{T}(\mathfrak{g})$ acts non-trivially on at most one generator per graded component in any irreducible module in $\mathcal{I}_{\text{fin}}^{\ast}$, a fact re-proven via current algebra techniques.
• The parametrization of isomorphism classes is unique up to twisting by one-dimensional $\mathcal{T}(\mathfrak{g})$-modules, and the orbit structure fully captures the classification.
• The proof technique avoids reliance on Heisenberg algebra representations from [F], instead using integral forms of $\mathcal{U}(\mathfrak{g}_{\text{aff}})$ and current algebra actions to derive the same classification result.