In this paper, we construct a homomorphism from the affine super Yangian $Y_{\ve_1,\ve_2}(\widehat{\mathfrak{sl}}(m|n))$ to the universal enveloping algebra of the rectangular $W$-superalgebra $W^{k}(\mathfrak{gl}(ml|nl),(l^{(m|n)}))$. We also show that the image of this homomorphism is dense provided that $k+(m-n)(l-1)\neq0$.
We classify irreducible representations of finite $W$-algebra of the queer Lie superalgebra $Q(n)$ associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of simple finite-dimensional representations of the super Yangian $YQ(1)$.
We introduce super Yangians of $\mathfrak{gl}(V),\mathfrak{sl}(V)$ (in the new Drinfeld realization) associated to all Dynkin diagrams of $\mathfrak{gl}(V)$, where $V$ is a finite-dimensional super vector space. We show that all of them are isomorphic to the super Yangians introduced by Maxim Nazarov, by identifying them with the corresponding RTT super Yangians. However, their "positive halves" are not pairwise isomorphic, and we obtain the shuffle algebra realizations of those. We adapt the latter to the trigonometric setup by obtaining the shuffle algebra realizations of the "positive halve…
Let $e$ be an arbitrary even nilpotent element in the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$ and let $\mathcal{W}_e$ be the associated finite $W$-superalgebra. Let $Y_{m|n}$ be the super Yangian associated to the Lie superalgebra $\mathfrak{gl}_{m|n}$. A subalgebra of $Y_{m|n}$, called the shifted super Yangian and denoted by $Y_{m|n}(σ)$, is defined and studied. Moreover, an explicit isomorphism between $\mathcal{W}_e$ and a quotient of $Y_{m|n}(σ)$ is established.
We study the finite W-superalgebra $W_e$ associated to a nilpotent element $e$ in a general linear Lie superalgebra. Under certain restriction on the Jordan type of $e$, we give a realization of $W_e$ in terms of a quotient of a shifted super Yangian.
Consider the complex matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism $η$ of $gl_{N|N}$ by $η(E_{ij})=E_{-i,-j}$. The queer Lie superalgebra $q_N$ is the fixed point subalgebra in $gl_{N|N}$ relative to the automorphism $η$. Consider the twisted polynomial current Lie superalgebra $g=\{X(t)\in\gl_{N|N}[t]:η(X(t))=X(-t)\}$. The enveloping algebra $U(g)$ of the Lie superalgebra $g$ has a deformation, called the Yangian of $q_N$. For each $M=1,2,...$ denote by $A_N^M$ the centralizer of $q_M\subset q_{N+M}$ i…
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 show that parabolic inductions for representations of the rectangular $W$-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super setting.
Take the matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism of $gl_{N|N}$ by sending $E_{ij}$ to $E_{-i,-j}$. Then the corresponding twisted subalgebra $g$ in the polynomial current Lie superalgebra $gl_{N|N}[u]$, has a natural Lie co-superalgebra structure. Here we quantise the universal enveloping algebra $U(g)$ as a co-Poisson Hopf superalgebra. For the quantised algebra we give a description of the centre, and construct the double in the sense of Drinfeld. We also construct a class of irreducible represe…
The present paper is devoted to studying the super Yangian $Y_{m|n}$ associated to the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ over a field of positive characteristic. We extend Drinfeld-type presentations of $Y_{m|n}$ and the special super Yangian $SY_{m|n}$ to positive characteristic. Moreover, the center $Z(Y_{m|n})$ of $Y_{m|n}$ is described: it is generated by its Harish-Chandra center together with a large $p$-center. We also study the $p$-center of $SY_{m|n}$ and provide another description of the $p$-center of $Y_{m|n}$ in terms of the RTT generators.
Associated to a composition of M and a composition of N, a new presentation of the super Yangian of the general linear Lie superalgebra $Y(gl_{M|N})$ is obtained.
A homomorphism from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))$
In order to extend the geometrization of Yangian $R$-matrices from Lie algebras $gl(n)$ to superalgebras $gl(M|N)$, we introduce new quiver-related varieties which are associated with representations of $gl(M|N)$. In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre tran…
Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$
We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called $q$-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of F…
