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 a minimalistic presentation of Drinfeld super Yangians in the case of special linear superalgebra associated with an arbitrary Dynkin diagram. This gives us a possibility to introduce Hopf superalgebra structure on Drinfeld super Yangians. Using complete Weyl group we classify Drinfeld super Yangians endowed with mentioned Hopf superalgebra structures. Also it is constructed an isomorphism between completions of Drinfeld super Yangians and quantum loop superalgebras.
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…
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 $U_{\hbar}(LA(m,n))$.
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.
We give a new proof of the fact that the super Yangian of general linear Lie superalgebra is isomorphic to the finite W-superalgebra of the general linear Lie superalgebra associated to a rectangular nilpotent element.
Yangian Double $DY(A(m,n))$ of Lie Superalgebra $A(m,n)$ is described in terms of generators and defining relations. It is proved triangular decomposition for Yangian $Y(A(m,n))$ and its quantum double $DY(A(m,n))$ as a corollary of PBW theorem. It is introduced the normally ordered bases in Yangian and its dual Hopf superalgebra in quantum double. It is calculated the pairing formulas between the elements of its bases. It is received the formula of Universal $R$-matrix of Yangian Double. The formula for Universal $R$-matrix of Yangian, which was introduced by V. Drinfel'd, is received also.
We show that some finite W-superalgebras based on gl(M|N) are truncation of the super-Yangian Y(gl(M|N)). In the same way, we prove that finite W-superalgebras based on osp(M|2n) are truncation of the twisted super-Yangians Y(gl(M|2n))^{+}. Using this homomorphism, we present these W-superalgebras in an R-matrix formalism, and we classify their finite-dimensional irreducible representations.
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.
The centrally extended su(2|2) superalgebra is an asymptotic symmetry of the light-cone string sigma model on AdS5 x S5. We consider an evaluation representation of the conventional Yangian built over a particular 16-dimensional long representation of the centrally extended su(2|2). Interestingly, we find that S-matrices compatible with this evaluation representation do not exist. On the other hand, by requiring centrally extended su(2|2) invariance and explicitly solving the Yang-Baxter equation, we find a scattering matrix for long-short representations of the Lie superalgebra. We notice tha…
Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$, II: Quantum currents
Dipartimento di Matematica, Sapienza Università di Roma（罗马大学数学系）；INFN sezione di Roma（罗马意大利国家核物理研究所）；University of Zagreb Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）
Quantum Yangian symmetry in several sigma models with supergroup or supercoset as target is established. Starting with a two-dimensional conformal field theory that has current symmetry of a Lie superalgebra with vanishing Killing form we construct non-local charges and compute their properties. Yangian axioms are satisfied, except that the Serre relations only hold for a subsector of the space of fields. Yangian symmetry implies that correlation functions of fields in this sector satisfy Ward identities. We then show that this symmetry is preserved by certain perturbations of the conformal fi…
Yang-Baxter relations symmetric with respect to the ortho-symplectic superalgebras are studied. We start from the formulation of graded algebras and the linear superspace carrying the vector (fundamental) representation of the ortho-symplectic supergroup. On this basis we study the analogy of the Yang-Baxter operators considered earlier for the cases of orthogonal and symplectic symmetries: the vector (fundamental) R matrix, the L operator defining the Yangian algebra and its first and second order evaluations. We investigate the condition for L(u) in the case of the truncated expansion in inv…
We describe a Gauss decomposition for the Yangian Y(gl_{m|n}) of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m-1,n-1)) (with m and n not equal) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians Y(sl_{n|n}) and Y(A(n,n)). We also show that the coefficients of the quantum Berezinian generate the centre of Y(gl_{m|n}). This was conjectured by Nazarov in 1991.
We construct highest-weight modules and a Yangian extension of the centrally extended sl(1|1)^2 superalgebra, that is a symmetry of the worldsheet scattering associated with the AdS3/CFT3 duality. We demonstrate that the R-matrix intertwining atypical modules is of a trigonometric type. We also consider a quantum deformation of this superalgebra, its modules, and obtain a quantum affine extension of the Drinfeld-Jimbo type that describes a deformed worldsheet scattering.
We construct universal Drinfel'd twists defining deformations of Hopf algebra structures based upon simple Lie algebras and contragredient simple Lie superalgebras. In particular, we obtain deformed and dynamical double Yangians. Some explicit realisations as evaluation representations are given for sl(N), sl(1|2) and osp(1|2).
For any fixed composition $μ$ of $M+N$ and any fixed $0^M1^N$-sequence $\mathfrak{s}$, we obtain a new presentation of the super Yangian $Y_{M|N}$ associated to the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$.
The odd reflections are an effective tool in the Lie superalgebra representation theory, as they relate non-conjugate Borel subalgebras. We introduce analogues of the odd reflections for the Yangian ${\rm Y}({\frak gl}_{m|n})$ and use them to produce a transition rule for the parameters of the highest weight modules corresponding to a change of the parity sequence. This leads to a description of the finite-dimensional irreducible representations of the Yangian associated with an arbitrary parity sequence.
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…
Methods are developed for systematically constructing the finite dimensional irreducible representations of the super Yangian Y(gl(M|N)) associated with the Lie superalgebra gl(M|N). It is also shown that every finite dimensional irreducible representation of Y(gl(M|N)) is of highest weight type, and is uniquely characterized by a highest weight. The necessary and sufficient conditions for an irrep to be finite dimensional are given.
Brundan and Kleshchev recently introduced a new family of presentations of the Yangian Y(gl_n) associated to the general linear Lie algebra gl_n, and thus provided a fresh approach to its study. In this article, we show how some of their ideas can be fruitfully extended to consider the Yangian Y(gl_{m|n}) associated to the Lie superalgebra gl_{m|n}. In particular, we give a new proof of the result by Nazarov that the quantum Berezinian is central.
We describe the realization of the super Reshetikhin-Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel-Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between the super RS algebra and the Drinfeld current realization of quantum affine superalgebras is established by using the Gauss decomposition technique of Ding-Frenkel. As an application, we obtain Drinfeld realization of quantum affine superalgebra $U_q[osp(1|2)^{(1)}]$ and its degeneration -- central extended super Yangian double $DY_\hbar[osp(1|2)^…
We discuss the integrability structure of deformed, four-dimensional N=4 super Yang-Mills theories using Yangians. We employ a recent procedure by Beisert and Roiban that generalizes the beta deformation of Lunin and Maldacena to produce N=1 superconformal gauge theories, which have the superalgebra SU(2,2|1)xU(1)xU(1). The deformed theories, including those with the more general twist, were shown to have retained their integrable structure. Here we examine the Yangian algebra of these deformed theories. In a five field subsector, we compute the two cases of SU(2)xU(1)xU(1)xU(1) and SU(2|1)xU(…
Covariant tensor representations of gl(m|n) occur as irreducible components of tensor powers of the natural (m+n)-dimensional representation. We construct a basis of each covariant representation and give explicit formulas for the action of the generators of gl(m|n) in this basis. The basis has the property that the natural Lie subalgebras gl(m) and gl(n) act by the classical Gelfand-Tsetlin formulas. The main role in the construction is played by the fact that the subspace of gl(m)-highest vectors in any finite-dimensional irreducible representation of gl(m|n) carries a structure of an irredu…
Representations of Super Yangians with Gelfand-Tsetlin bases
Shenzhen International Center for Mathematics, Southern University of Science and Technology（深圳数学中心，南方科技大学）；School of Artificial Intelligence, Jianghan University（江汉大学人工智能学院）；School of Mathematics and Statistics, Central China Normal University（华中师范大学数学与统计学院）
研究超杨代数的表示，通过评价同态将其协变张量模视为超杨代数模，并给出简单商模块的 tame 条件，扩展了纳扎罗夫和塔拉索夫的工作。
We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb C)$. Our main result gives an explicit description of $W_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y(\mathfrak{gl}_{1|1})$. We also show that $W_{m|n}$ admits a triangular decomposition and construct its irreducible representations.
Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1}$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ a minimal nilpotent element in $\mathfrak{g}_{\bar 0}$. Set $W_χ'$ to be the refined $W$-superalgebra associated with the pair $(\mathfrak{g},e)$, which is called a minimal $W$-superalgebra. In this paper we present a set of explicit generators of minimal $W$-superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$, the lower bounds of dimensions in the modular representations of ba…
We use the super version of RS construction method and Gauss decomposition to obtain Drinfeld's currents realization of super Yangian double $DY(gl(1|1))$ with center extension
We introduce super Yangian double $DY_\hbar[gl(m|n)]$ and its central extension $\widehat{DY_\hbar[gl(m|n)]}$. We give their defining relations in terms of current generators and obtain Drinfeld comultiplication.
