[Paper Review] Yangian actions on higher level irreducible modules of affine gl(N)

Denis Uglov|arXiv (Cornell University)|Feb 9, 1998

Advanced Algebra and Geometry3 citations

TL;DR

This paper constructs an action of the Yangian of gl(N) on irreducible highest weight modules of affine gl(N) at integer level >1, using Drinfeld duality and a semi-infinite limit from a degenerate double affine Hecke algebra (H) representation. The vacuum modules decompose into irreducible Yangian subrepresentations, parameterized by semi-infinite skew Young diagrams, providing a new realization of these modules via H-intertwiners.

ABSTRACT

An action of the Yangian of the general Lie algebra gl(N) is defined on every irreducible highest weight module of affine gl(N) with integer level greater than 1. This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra H. Each vacuum module of affine gl(N) is decomposed into irreducible Yangian subrepresentations by means of the intertwiners of H. Components of this decomposition are parameterized by semi-infinite skew Young diagrams.

Motivation & Objective

• To define a Yangian action on irreducible highest weight modules of affine gl(N) at integer level greater than 1.
• To establish a connection between the degenerate double affine Hecke algebra H and the representation theory of affine gl(N) via a semi-infinite limit construction.
• To decompose vacuum modules of affine gl(N) into irreducible Yangian subrepresentations using intertwiners from H.
• To parameterize the components of this decomposition using semi-infinite skew Young diagrams.
• To provide a new algebraic framework for understanding higher level representations of affine gl(N) through Yangian symmetries.

Proposed method

• The construction begins with an induced representation of the degenerate double affine Hecke algebra H.
• Drinfeld duality is applied to relate H-representations to Yangian representations of gl(N).
• A semi-infinite limit is taken to descend from the H-representation to the affine gl(N) setting.
• Intertwiners of H are used to decompose vacuum modules of affine gl(N) into irreducible Yangian subrepresentations.
• The resulting decomposition is indexed by semi-infinite skew Young diagrams, which classify the components.

Experimental results

Research questions

• RQ1 How can the Yangian of gl(N) be realized as an action on irreducible highest weight modules of affine gl(N) at level >1?
• RQ2 What role does the degenerate double affine Hecke algebra H play in constructing this Yangian action?
• RQ3 How do the intertwiners of H facilitate the decomposition of vacuum modules into irreducible Yangian subrepresentations?
• RQ4 What combinatorial structure parameterizes the components of this decomposition?
• RQ5 In what way does the semi-infinite limit construction bridge H-representations and affine gl(N) modules?

Key findings

• The Yangian of gl(N) acts on every irreducible highest weight module of affine gl(N) at integer level greater than 1.
• This action is constructed via Drinfeld duality and a semi-infinite limit from a representation of the degenerate double affine Hecke algebra H.
• Vacuum modules of affine gl(N) decompose into irreducible Yangian subrepresentations through H-intertwiners.
• The components of this decomposition are classified by semi-infinite skew Young diagrams.
• The construction provides a new realization of higher level affine gl(N) representations using Yangian symmetry and combinatorial indexing.