[Paper Review] Mickelsson algebras and representations of Yangians

S. Khoroshkin, Maxim Nazarov|arXiv (Cornell University)|Dec 6, 2009

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper constructs irreducible finite-dimensional representations of twisted Yangians Y(spₙ) and Y(soₙ) using quotients of tensor products of symmetric and exterior powers of ℂⁿ. It provides new realizations of these modules via Mickelsson algebras and extremal projectors, establishing a correspondence between irreducible modules and sequences of monic polynomials, with splitting conditions tied to weight singularities and Weyl group actions via Zhelobenko automorphisms.

ABSTRACT

We use the theory of reductive dual pairs due to Howe to obtain explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra, and of the twisted Yangians corresponding to the symplectic and orthogonal Lie algebras.

Motivation & Objective

• To provide new realizations of irreducible finite-dimensional modules for the twisted Yangians Y(spₙ) and Y(soₙ) using tensor product quotients of symmetric and exterior powers of ℂⁿ.
• To extend known realizations for Y(glₙ) to twisted Yangians using Mickelsson algebra techniques and extremal projectors.
• To characterize irreducible modules of Y(soₙ) via polynomial invariants and Weyl group actions, under the condition that the soₙ-action integrates to SOₙ.
• To establish a correspondence between irreducible modules and sequences of monic polynomials, with splitting behavior determined by weight singularities.
• To clarify the role of Zhelobenko automorphisms and the dynamical Weyl group in the representation theory of twisted Yangians.

Proposed method

• Constructs the Mickelsson algebra R as a quotient of the normalizer of a right ideal J = nA in an associative algebra A containing U(g), using the adjoint action of a reductive subalgebra g.
• Introduces the ring of fractions A̅ of A and defines R̅ as a quotient of the normalizer of nA̅, which admits a bijective description via the quotient space Z̅ = A̅/(J̅ + J̅′) where J̅′ is the opposite nilpotent ideal.
• Uses the extremal projector for g to describe the multiplication in R̅ in terms of the vector space Z̅, enabling explicit module constructions.
• Applies the Zhelobenko automorphisms—induced by the braid group action on Z̅—to define Weyl group actions on the subalgebra of zero-weight elements in Z̅.
• Leverages the G-action on A (extending the adjoint action of G on U(g)) to define G-invariant elements and project them onto Z̅ to construct modules.
• Constructs intertwining operators between modules via the action of the Yangian, with kernel quotients yielding irreducible representations, and determines splitting via the label δ = ±1.

Experimental results

Research questions

• RQ1 How can irreducible finite-dimensional modules of the twisted Yangians Y(spₙ) and Y(soₙ) be realized as quotients of tensor products of symmetric and exterior powers of ℂⁿ?
• RQ2 What is the role of Mickelsson algebras and extremal projectors in constructing and classifying representations of twisted Yangians?
• RQ3 How do Zhelobenko automorphisms and the Weyl group action on the zero-weight subalgebra of Z̅ relate to the representation theory of Y(soₙ)?
• RQ4 Under what conditions does the intertwining operator between Y(soₙ)-modules split into two non-equivalent irreducible components?
• RQ5 What polynomial invariants classify irreducible finite-dimensional Y(soₙ)-modules, and how are they related to the weights λ and μ?

Key findings

• All irreducible finite-dimensional modules of Y(spₙ) are realized as quotients of tensor products of symmetric and exterior powers of ℂⁿ.
• For Y(soₙ), all irreducible finite-dimensional modules whose soₙ-action integrates to SOₙ arise as quotients of such tensor products, with the condition that λ + ρ is nonsingular.
• The intertwining operator (4.58) yields an irreducible Y(soₙ)-module if the kernel quotient is irreducible, or splits into two non-equivalent irreducible modules if δ = ±1.
• The polynomial Qₗ(x) associated with a module is the product of differences x² − (μₐ + ρₐ)² over indices a with νₐ = l, and splitting occurs iff μₐ + ρₐ = 0 for some a with νₐ = l.
• The weights λ and μ are determined up to the shifted Weyl group action of the group R of so₂ₘ, with the freedom to permute pairs (λₐ + ρₐ, μₐ + ρₐ) and apply sign flips.
• When ν₁ = … = νₘ = 0, both source and target modules in (4.58) are trivial, and the operator becomes the identity map ℂ → ℂ if λ + ρ is nonsingular.