Induced Modules for Affine Lie Algebras (0810.3458v2)

Abstract: We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra ${\mathcal P}$ of an affine Lie algebra ${\mathfrak G}$, our main result establishes the equivalence between a certain category of ${\mathcal P}$-induced ${\mathfrak G}$-modules and the category of weight ${\mathcal P}$-modules with injective action of the central element of ${\mathfrak G}$. In particular, the induction functor preserves irreducible modules. If ${\mathcal P}$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra ${\mathcal P}^{ps}$, ${\mathcal P}\subset {\mathcal P}^{ps}$. The structure of ${\mathcal P}$-induced modules in this case is fully determined by the structure of ${\mathcal P}^{ps}$-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. K\"onig, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63].

• The paper establishes an equivalence between categories of P-induced modules and weight modules, preserving irreducibility under the induction functor.
• It generalizes earlier results by introducing pseudo parabolic subalgebras to unify the structure theory of modules for affine Lie algebras.
• It provides detailed conditions for module irreducibility with nonzero central charge, advancing classification in representation theory.

Overview of "Induced Modules for Affine Lie Algebras" by Vyacheslav Futorny and Iryna Kashuba

This paper investigates the theory of induced modules over affine Lie algebras, which are a subclass of Kac-Moody algebras. Affine Lie algebras play a crucial role in mathematical physics, representation theory, and related fields due to their structural complexity and numerous applications. The authors, Vyacheslav Futorny and Iryna Kashuba, extend the theory of induced modules by introducing a general reduction theorem that applies to modules with nonzero central charge.

Main Contributions

1. Equivalence of Categories: The paper's main theorem establishes an equivalence between certain categories of $\mathcal{P}$-induced $\mathfrak{G}$-modules, where $\mathcal{P}$ is a pseudo parabolic subalgebra of an affine Lie algebra $\mathfrak{G}$, and weight $\mathcal{P}$-modules. This equivalence is significant because it preserves irreducible modules under the induction functor.
2. Generalization of Previous Results: The presented results generalize previous work on reductions in specific cases, as noted by Futorny and colleagues in earlier papers [Futorny, K\"onig, Mazorchuk (2001), Cox (1994), Dimitrov, Futorny, Penkov (2004)]. By introducing pseudo parabolic subalgebras and their properties, the authors provide a unifying framework that subsumes these earlier findings.
3. Structure of Induced Modules: The paper meticulously develops the structure theory for $\mathcal{P}$-induced modules when $\mathcal{P}$ has a finite-dimensional Levi factor. The authors highlight that the structure of $\mathcal{P}$-induced modules can be fully determined by the structure of modules induced from a related unique pseudo parabolic subalgebra $\mathcal{P}^{ps}$.

Key Findings and Implications

• Reduction for Nonzero Central Charge: The paper's main reduction theorem reveals how unitary modules for affine algebras with nonzero central charge can be reduced to modules over proper subalgebras, thereby classifying irreducible modules. This classifies irreducible modules into those that can be obtained from parabolic induction and cuspidal modules.
• Cuspidal and Non-Cuspidal Modules: The treatment includes both cuspidal and non-cuspidal cases, with a focus on weight spaces. The paper emphasizes that understanding cuspidal modules is crucial for a complete classification.
• Irreducibility Conditions: Through the proposed framework, the authors establish conditions under which induced modules are irreducible. The equivalence of categories $W(\mathfrak{k})$ and $\tilde{\mathcal{O}}(\mathfrak{g}, \mathfrak{p})$ aids in mapping submodule structures, which has further theoretical importance in representation theory.

Future Directions

The authors' work opens several avenues for future research in representation theory for affine Lie algebras:

• Extending to Other Classes: Future research could explore extending the reduction theorems and category equivalences to other infinite-dimensional Lie algebras beyond the affine class.
• Cuspidal Module Classification: There is an opportunity for more in-depth classification of cuspidal modules, particularly in the context of nontrivial central extensions, which would benefit from the framework discussed in this paper.
• Applications to Mathematical Physics: Given the affinity of affine Lie algebras to integrable systems in physics, the theoretical developments here could lead to new insights or applications in domains like statistical mechanics or conformal field theory.

In conclusion, this paper makes significant strides in the paper of induced modules over affine Lie algebras, offering a generalized and comprehensive view of module structures and their irreducibility. The research lays a foundation for further exploration both within the theory of Kac-Moody algebras and in its applications across mathematics and physics.