R-matrix via Hasse diagrams
We calculate the R-matrix for the exceptional Lie superalgebra $\mathfrak{d}(2,1;κ)$ in the smallest representation of its quantum supergroup, using a method of Hasse diagrams.
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arXiv subjects
Vladimir Stukopin
Publications and source records attributed to Vladimir Stukopin.
Affine supersymmetric pairs
We classify Satake diagrams for general linear and orthosymplectic non-twisted affine Lie superalgebras and prove that each of them generates a family of proper spherical subalgebras. Furthermore, we demonstrate that every such family contains subalgebras with matrix invariants, which are viewed as classical analogs of K-matrices solving supersymmetric Reflection equation.
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Schubert varieties for the super affine Grassmannian of $GL_{n|m}$
We study Schubert varieties of the affine Grassmannian for the general linear supergroup $GL_{n|m}$. An explicit computational study is conducted in low-dimensional cases, namely for dimensions $n|m = 1|1$ and $2|1$. We describe the supervariety structures that arise in these settings, providing coordinate descriptions, equations, and morphisms.
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Coproducts for affine super-Yangian and Weyl groupoid action
For affine special linear superalgebra $\widehat{sl}(m|n, Π)$ defined by an arbitrary system of simple roots $Π$ we define the affine super Yangian $Y_{\hbar}(\widehat{sl}(m|n, Π))$ as Hopf superalgebra which is a quantization of superbialgebra $\widehat{sl}(m|n, Π)[t]$ and describe super Yangian in terms of minimalistic system of generators. We consider Drinfeld presentation for $Y^D_{\hbar}(\widehat{sl}(m|n, Π))$ and prove that these two presentations are isomorphic as associative superalgebras. We induce by means of this isomorphism a co-multiplication on the Drinfeld presentation $Y^D_{\hbar}(\widehat{sl}(m|n, Π))$ of the super Yangian. We introduce the action of Weyl groupoid by isomorphisms on super Yangians as an extension of its action on universal enveloping algebra and deformation of action on univesal enveloping superalgebra of current Lie superalgebra and prove that such extension exists and unique. As a consequence of this construction we obtain that super Yangians $Y_{\hbar}(\widehat{sl}(m|n, Π_1))$ and $Y_{\hbar}(\widehat{sl}(m|n, Π_2))$, defined by different simple root systems $ Π_1$ and $ Π_2$ are isomorphic as Hopf superalgebras.
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Mickelsson algebras and inverse Shapovalov form
Let $\mathcal{A}$ be an associative algebra containing the classical or quantum universal enveloping algebra $U$ of a semi-simple complex Lie algebra. Let $\mathcal{J}\subset \mathcal{A}$ designate the left ideal generated by positive root vectors in $U$. We construct the reduction algebra of the pair $(\mathcal{A},\mathcal{J})$ via the inverse Shapovalov form of $U$.
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Mickelsson algebras via Hasse diagrams
Let $\mathcal{A}$ be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra $\mathfrak{g}$. We present a construction of the Mickelsson algebra $Z(\mathcal{A},\mathfrak{g})$ relative to the left ideal in $\mathcal{A}$ generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum $\mathfrak{g}$-modules. We give an explicit expression for a PBW basis in $Z(\mathcal{A},\mathfrak{g})$ in the case when $\mathcal{A}=U(\mathfrak{a})$ of a finite-dimensional Lie algebra $\mathfrak{a}\supset \mathfrak{g}$. For $\mathcal{A}=U_q(\mathfrak{a})$ and $\mathfrak{g}$ the commutant of a Levi subalgebra in $\mathfrak{a}$, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to $\mathbb{C}[[\hbar]]$.
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Affine Super Yangian and Weyl groupoid
We define affine Super Yangian $Y_{\hbar}(\hat{sl}(m|n), Π) $ for affine special linear superalgebra $\hat{sl}(m|n)$ and arbitrary system of simple roots $Π$ in terms of minimalistic system of generators. We also consider Drinfeld presentation for affine super Yangian in the case of arbitrary simple root system $Π$ and prove that these two presentations (Drinfeld and minimalistic) of $Y_{\hbar}(\hat{sl}(m|n), Π)$ are isomorphic as associative superalgebras. We also construct isomorphism of affine super Yangians $Y_{\hbar}(\hat{sl}(m|n), Π)$ and $Y_{\hbar}(\hat{sl}(m|n), Π')$ for different simple root systems $Π$ and $Π'$. After them we also define Weyl groupoid as a set of morphisms in category with objects, which are super Yanginas $Y_{\hbar}(\hat{sl}(m|n), Π)$, where $Π$ is simple root system. We describe Weyl groupoid in terms of generators and describe action of these generators on super Yangians. We describe isomorphisms between $Y_{\hbar}(\hat{sl}(m|n), Π)$ and $Y_{\hbar}(\hat{sl}(m|n), Π')$ as elements of Weyl groupoid.
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Drinfeld Yangian of the queer Lie superalgebra. I
Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case Lie superalgebra $sq_1$ that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev-Reshetikhin-Takhtadzhjan approach, are isomorphic.
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Isomorphism between super Yangian and quantum loop superalgebra. I
Following V. Toledano-Laredo and S. Gautam approach we construct isomorphism between super $\hbar$-Yangian $Y_{\hbar}(A(m,n))$ of special linear superalgebra and quantum loop superalgebra $U_{\hbar}(LA(m,n))$.
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Yangian of the Strange Lie Superalgebra of $\boldsymbol{Q_{n-1}}$ Type, Drinfel'd Approach
The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described.
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