2-Kac-Moody algebras (0812.5023v1)

Abstract: We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations.

• The paper introduces a novel 2-categorical construction for Kac-Moody algebras that enhances 2-representation theory.
• It refines methods using symmetric and Hecke algebras to establish key correspondences in graded and nilpotent categories.
• The work lays a foundation for applying these insights to topological quantum field theories and broader categorification efforts.

Overview of "2-Kac-Moody Algebras" by Raphael Rouquier

The paper by Raphael Rouquier addresses the construction and exploration of 2-categories associated with Kac-Moody algebras, particularly focusing on their 2-representations. This work extends the frameworks for categorification initially applied to simpler structures, aiming to provide a coherent pathway for enhancing the interaction and representation theory of Kac-Moody algebras.

Key Contributions

The primary contribution is the development of a 2-category, denoted as $\mathcal{A}(g)$, linked to a Kac-Moody algebra $g$. This framework systematically enhances the paper of such algebras by introducing new perspectives on their representations, effectively dealing with higher category structures. The approach leverages existing methodologies and extends them to introduce 2-representations, drawing from monoidal categories and their associated properties.

Structure and Components

• 2-Categorical Framework: The paper refines the concept of 2-categories associated with Kac-Moody algebras, providing a detailed treatment underpinned by generators and relations. This setup allows for the introduction of a categorical structure embedded with natural isomorphisms that facilitate map compositions, effectively extending the representation theory from one category to two categories.
• Symmetric Algebras and Hecke Algebras: Rouquier elaborates on symmetric algebras' role, emphasizing their representation via Hecke algebras—classical and degenerate forms—and their associated polynomial realizations. This segment underscores the generation of nilpotent categories in the context of hermitian matrices, which plays a pivotal role in the structure of nil Hecke algebras.
• 2-Reprepresentations and Categorifications: The paper transitions to explore the structure of 2-representations, particularly exploring their integrability. Rouquier discusses methodologies for constructing these representations and examines their relation to classical algebraic structures like the Hecke algebra. Here, the paper articulates the equivalence between categorifications of $\text{sl}_2$ and integrable 2-representations, setting a foundation for establishing 2-functor actions through specific transformations.

Numerical and Conceptual Highlights

• Braid Group Actions and Equivalences: The paper shows that the braid group actions maintain equivalence, a critical element for ensuring that this high-level categorification remains consistent with lower-dimension phenomena.
• Graded Categories and Specializations: Detailed discussions about graded categories provide insight into the transformations possible through component specializations. The paper especially highlights the uniqueness introduced by a Kac-Moody algebra's structure and its significant influence when transposing classical forms to graded frameworks.

Implications and Future Directions

Practically, this research opens pathways for creating topological quantum field theories (TQFTs) with enhanced dimensionality, as posited in the Crane-Frenkel problem discussed in the paper. Theoretically, Rouquier's extension of representation theory to 2-categories adds a crucial layer of abstraction, potentially influencing future studies in non-commutative geometry and related areas such as representation theory of quivers and algebraic geometry.

Looking ahead, further investigations into broader Kac-Moody contexts, such as affine and hyperbolic types, may reveal additional dimensions to these structures. Furthermore, elaborating upon 2-categorifications and their practical implementations in quantum algebra presents an intriguing avenue for continued inquiry within both mathematical contexts and applied sciences.

Rouquier's work sets a high standard for the rigor and breadth required in approaching complex algebraic structures through categorification, providing a solid foundation upon which future explorations can build.