Related papers: On Guay's evaluation map for affine Yangians

The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence…

We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we…

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible…

We study in detail the structure of the Yangian Y(gl(N)) and of some new Yangian-type algebras called twisted Yangians. The algebra Y(gl(N)) is a `quantum' deformation of the universal enveloping algebra U(gl(N)[x]), where gl(N)[x] is the…

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra U_h(Lg) of g degenerates to the Yangian Y_h(g). We strengthen this result by constructing an explicit algebra homomorphism Phi defined over Q[[h]]…

We construct Young wall models for the crystal bases of level $1$ irreducible highest weight representations and Fock space representations of quantum affine algebras in types $E_{6}^{(1)}$, $E_{7}^{(1)}$ and $E_{8}^{(1)}$. In each case,…

Let $\mathfrak g$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak g$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite…

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 $\lambda$ and $\mu$ for a Lie algebra $\mathfrak{g}$, which we will assume is…

We construct four edge contractions for the affine super Yangian of type $A$. By using these edge contractions, we give a homomorphism from the affine super Yangian of type $A$ to the universal enveloping algebra of the non-rectangular…

In this paper we construct bases of standard (i.e. integrable highest weight) modules $L(\Lambda)$ for affine Lie algebra of type $B_2\sp{(1)}$ consisting of semi-infinite monomials. The main technical ingredient is a construction of…

The affine Yangian of $\mathfrak{gl}_1$ has recently appeared simultaneously in the work of Maulik-Okounkov and Schiffmann-Vasserot in connection with the Alday-Gaiotto-Tachikawa conjecture. While the former presentation is purely…

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${\frak osp}_{2n+1|2m}$. Every such representation is highest weight and we use embedding theorems…

We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph's characteristic varieties. We show that an…

We define natural classes of rational and polynomial representations of the Yangian of the general linear Lie algebra. We also present the classification and explicit realizations of all irreducible rational representations of the Yangian.

For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the…

This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal…

In this paper, we define the affine super Yangian $Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}}(m|n))$ with a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from…

We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give…

We continue the study of finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types B, C and D, with focus on those of types BI, CII and DI. After establishing that, for all twisted Yangians of…

We prove several basic properties of the Yangian of the general linear Lie superalgebra.