Coproduct for Yangians of affine Kac–Moody algebras

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights λ and μ for a Lie algebra frakturg, which we will assume is simply laced. In this paper, we relate the category scriptO over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category scriptO is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category scriptO for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category scriptO are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand–Tsetlin modules of U(frakturgln) and its associated W‐algebras.

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“…It seems likely that by appropriately incorporating the full Cartan into the definition of

Y_{μ}

( c.f. [, Definition 2.1] ) , we could avoid having to choose between generators

A_{i}^{false(p false)}

and

H_{i}^{false(q false)}

.…”

Section: Truncated Shifted Yangians and Klr Yangian Algebrasmentioning

confidence: 99%

On category O for affine Grassmannian slices and categorified tensor products

Kamnitzer

Tingley

Webster

et al. 2019

Proc. London Math. Soc.

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“…In this paper, we defineŝl N = sl N ⊗ C t, t −1 ⊕ Cc without the degree operator. In [6], the Yangian Y (g) of affine type is defined to be an algebra containing the degree operator d and one without d is denoted by Y (g ). In particular, the algebra defined in Definition 2.1 coincides with Y ,ε (g ) in the notation of [6, Definition 7.1].…”

Section: Affine Yangianmentioning

confidence: 99%

“…A formula for the coproduct on the affine Yangian Y ŝ l N was stated in [5]. Guay-Nakajima-Wendlandt [6] gave a detailed proof of the well-definedness. To recall it, we consider a bigger algebra Y ŝ l N ⊕ Cd , which is generated by x ± i,r , h i,r , d with defining relations given in [6, equation (2.8)].…”

Section: Coproductmentioning

confidence: 99%

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Braid Group Action on Affine Yangian

Kodera

2019

SIGMA

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“…One way of seeing this is to appeal to the proof of [GRW4, Theorem 2.6]. A careful reading of that proof together with [GNW,3(ii)] shows that if g ≇ sl 2 then the relation (3.4) can be omitted and the relation (3.3) can even be replaced with the relation…”

Section: The Yangian Of a Simple Lie Algebramentioning

confidence: 99%

The R-Matrix Presentation for the Yangian of a Simple Lie Algebra

Wendlandt

2018

Commun. Math. Phys.

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Starting from a finite-dimensional representation of the Yangian Y (g) for a simple Lie algebra g in Drinfeld's original presentation, we construct a Hopf algebra X I (g), called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific R-matrix R(u). We prove that there is a surjective Hopf algebra morphism X I (g) ։ Y (g) whose kernel is generated as an ideal by the coefficients of a central matrix Z(u). When the underlying representation is irreducible, we show that this matrix becomes a grouplike central series, thereby making available a proof of a well-known theorem stated by Drinfeld in the 1980's. We then study in detail the algebraic structure of the extended Yangian, and prove several generalizations of results which are known to hold for Yangians associated to classical Lie algebras in their R-matrix presentations.

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Coproduct for Yangians of affine Kac–Moody algebras

Cited by 52 publications

References 37 publications

On category O for affine Grassmannian slices and categorified tensor products

On category O for affine Grassmannian slices and categorified tensor products

Braid Group Action on Affine Yangian

The R-Matrix Presentation for the Yangian of a Simple Lie Algebra

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