We give a presentation for the finite W-algebra associated to a nilpotent matrix inside the general linear Lie algebra over C. In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l, the presentation is that of the Yangian of level l associated to the Lie algebra gl_n, as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians.
We introduce a class of new algebras, the shifted quiver Yangians, as the BPS algebras for type IIA string theory on general toric Calabi-Yau three-folds. We construct representations of the shifted quiver Yangian from general subcrystals of the canonical crystal. We derive our results via equivariant localization for supersymmetric quiver quantum mechanics for various framed quivers, where the framings are determined by the shape of the subcrystals. Our results unify many known BPS state counting problems, including open BPS counting, non-compact D4-branes, and wall crossing phenomena, simply…
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.
Remarkable subalgebras of the Yangian for gl_n called the shifted Yangians were introduced in a recent work by Brundan and Kleshchev in relation to their study of finite W-algebras. In particular, in that work a classification of finite-dimensional irreducible representations of the shifted Yangians and the associated finite W-algebras was given. We construct a class of these representations in an explicit form via bases of Gelfand-Tsetlin type.
We study a coproduct in type A quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl_2. At the classical level this corresponds to the multiplication of scattering matrices of euclidean SU(2) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra.
Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighbouring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.
We study the truncated shifted Yangian $Y_{n,l}(σ)$ over an algebraically closed field $\mathbb{k}$ of characteristic $p > 0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g}, e)$ associated to a corresponding nilpotent element $e\in \mathfrak{g} = \mathfrak{gl}_N(\mathbb{k})$. We obtain an explicit description of the centre of $Y_{n,l}(σ)$, showing that it is generated by its Harish-Chandra centre and its $p$-centre. We define $Y_{n,l}^{[p]}(σ)$ to be the quotient of $Y_{n,l}(σ)$ by the ideal generated by the kernel of trivial character of its $p$-centre. Our main th…
We define the Drinfeld generators for $Y_3^+$, the twisted Yangian associated to the Lie algebra $\mathfrak{so}_3(\mathbb{C})$. This allows us to define shifted twisted Yangians, which are certain subalgebras of $Y_3^+$. We show that there are families of homomorphisms from the shifted twisted Yangians in $Y_3^+$ to the universal enveloping algebras of various orthogonal and symplectic Lie algebras, and we conjecture that the images of these homomorphisms are isomorphic to various finite $W$-algebras.
We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture al…
We derive some new presentations for the Yangian associated to the Lie algebra gl_n(C) that are adapted to parabolic subalgebras. At one extreme, the presentation is just the usual RTT presentation, whilst at the other extreme it is a variation on Drinfeld's presentation. All these presentations play an important role in our subsequent article "Shifted Yangians and finite W-algebras".
We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the $\mathcal{N}=2$ affine Yangian, which is isomorphic to the universal enveloping algebra of $\mathfrak{u}(1)\oplus \mathcal{W}^{\mathcal{N}=2}_{\infty}[λ]$. All algebras that we construct have natural …
We use the theory of reductive dual pairs due to Howe to obtain explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra, and of the twisted Yangians corresponding to the symplectic and orthogonal Lie algebras.
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.
Using vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the PBW property for simply laced affine Yangians.
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.
Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in Yanigian are discussed. As a corollary, similar relations are deduced for shifted Schur functions.
Given an affine Kac-Moody algebra, we explain how to construct a coproduct for its associated Yangian. In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of the Yangian when the Kac-Moody algebra is, more generally, symmetrizable.
For a large class of finite W algebras, the defining relations of a Yangian are proved to be satisfied. Therefore such finite W algebras appear as realisations of Yangians. This result is useful to determine properties of such W algebra representations.
Olshanski's centralizer construction provides a realization of the Yangian for the Lie algebra gl(n) as a subalgebra in the projective limit of a chain of centralizers in the universal enveloping algebras. We give a modified version of this construction based on a quantum analog of Sylvester's theorem. We then use it to get an algebra homomorphism from the Yangian to the transvector algebra associated with the general linear Lie algebras. The results are applied to identify the elementary representations of the Yangian by constructing their highest vectors explicitly in terms of elements of th…
We construct an algebra homomorphism between the Yangian Y(sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras.
We introduce an analogue of the composition of the Cherednik and Drinfeld functor for twisted Yangians. Our definition is based on the Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We show an algebra morphism between Yangians and some finite W-algebras. This correspondence is nicely illustrated in the framework of the Non Linear Schrodinger hierarchy. For such a purpose, we give an explicit realization of the Yangian generators in terms of deformed oscillators.
We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the boundary Yangian symmetry of an AdS/CFT superstring which ends on a boundary with non-trivial degrees of freedom and which preserves the full bulk Lie symmetry algebra. We also consider the spectrum of bulk and boundary states and some automorphisms of the underlying algebras.
We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.
Shifted twisted Yangians and affine Grassmannian islices
Shenzhen International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology（深圳国际数学中心及南方科技大学数学系）；Department of Mathematics, University of Virginia（弗吉尼亚大学数学系）；Département de mathématiques, Université de Sherbrooke（舍布鲁克大学数学系）
We construct an explicit set of generators for the finite W-algebras associated to nilpotent matrices in the symplectic or orthogonal Lie algebras whose Jordan blocks are all of the same size. We use these generators to show that such finite W-algebras are quotients of twisted Yangians.
We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik.
We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $\hat{\mathfrak{gl}}_N$.
We study the convolution algebra $H^{G\times \CC^{*}}_{*}(Z)$ of $G$-equivariant homology group on the Steinberg variety of type B/C and define an algebra $\widetilde{Y}$ that maps to $H^{G\times \CC^{*}}_{*}(Z)$. The Drinfeld new realization of the twisted Yangian associated to symmetric pairs is a quotient of $\widetilde{Y}$. We also study the $G$-equivariant case and prove that the twisted Yangian is the deformation of the twisted current algebra.
We study the Yangians Y(a) associated with the simple Lie algebras a of type B, C or D. The algebra Y(a) can be regarded as a quotient of the extended Yangian X(a) whose defining relations are written in an R-matrix form. In this paper we are concerned with the algebraic structure and representations of the algebra X(a). We prove an analog of the Poincare-Birkhoff-Witt theorem for X(a) and show that the Yangian Y(a) can be realized as a subalgebra of X(a). Furthermore, we give an independent proof of the classification theorem for the finite-dimensional irreducible representations of X(a) whic…
We prove how the Yangian of $\mathfrak{gl}_N$ in its RTT presentation and Olshanski's twisted Yangians for the orthogonal and symplectic Lie algebras can be obtained by a degeneration process from the corresponding quantum loop algebra and some of its twisted analogues.
The Yangian characters (or q-characters) are known to be closely related to the classical W-algebras and to the centers of the affine vertex algebras at the critical level. We make this relationship more explicit by producing families of generators of the W-algebras from the characters of the Kirillov-Reshetikhin modules associated with multiples of the first fundamental weight in types B and D and of the fundamental modules in type C. We also give an independent derivation of the character formulas for these representations in the context of the RTT presentation of the Yangians. In all cases …
We study a composition of two functors. The first one, from the category of modules over the Lie algebra $\gl_m$ to the category of modules over the degenerate affine Hecke algebra of $GL_N$, was introduced by I. Cherednik. The second functor is a skew version of the functor from the latter category to the category of modules over the Yangian $\Y(\gl_n)$ due to V. Drinfeld. We give a representation theoretic explanation of a link between intertwining operators on tensor products of $\Y(\gl_n)$-modules, and the `extremal cocycle' on the Weyl group of $\gl_m$ introduced by D. Zhelobenko. We esta…
We study the composition of the functor from the category of modules over the Lie algebra gl_m to the category of modules over the degenerate affine Hecke algebra of GL_N introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian Y(gl_n) due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the tensor products of Y(gl_n)-modules, and the `extremal cocycle' on the Weyl group of gl_m defined by D. Zhelobenko. We also establish a connection between the composition of two functors…
The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem for the quantum affine algebra and for the twisted q…
Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL(N) and the Yangian for gl(N). We prove a version of this theorem for the twisted Yangians Y(g(N)) associated with the orthogonal and symplectic Lie algebras g(N)=o(N) or sp(N). This gives rise to representations of the twisted Yangian Y(g(N-M)) on the space of g(M)-homomorphisms Hom(W,V), where W and V are finite-dimensional irreducible modules over g(M) and g(N), respectively. In the symplectic case these representations turn out to…
We study a category O of representations of the Yangian associated to an arbitrary finite-dimensional complex simple Lie algebra. We obtain asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov--Reshetikhin modules. In the Grothendieck ring we establish the three-term Baxter's TQ relations for the asymptotic modules. We indicate that Hernandez--Jimbo's limit construction can also be applied, resulting in modules over anti-dominantly shifted Yangians.
We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.
We prove the equivalence of two presentations of the Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ and we also show the equivalence with a third presentation when $\mathfrak{g}$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.
We define the twisted affine Yangian of type $C$ and construct surjective homomorphisms from twisted affine Yangians of type $C$ to the universal enveloping algebra of the rectangular $W$-algebra associated with $\mathfrak{so}(ln)$ and a nilpotent element whose Jordan form corresponds to the partition $(l^n)$ in the case when $l$ and $n$ are even.
A general scheme of construction of Drinfeldians and Yangians from quantum non-twisted affine Kac-Moody algebras is presented. Explicit description of Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C, D are given in terms of a Cavalley basis.
We show that the truncation of twisted Yangians are isomorphic to finite W-algebras based on orthogonal or symplectic algebras. This isomorphism allows us to classify all the finite dimensional irreducible representations of the quoted W-algebras. We also give an R-matrix for these W-algebras, and determine their center.
We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algebra.
We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan's classification. These algebras can be regarded as coideal subalgebras of the extended Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincare-Birkoff-Witt Theorem, determine their centres and study also extended reflection algebras.
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with associated Yangian $Y_\hbar\mathfrak{g}$ and Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in \mathbb{C}$, there is an automorphism $τ_c$ of $Y_\hbar\mathfrak{g}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $τ_z$ from $Y_\hbar\mathfrak{g}$ to the polynomial algebra $Y_\hbar\mathfrak{g}[z]$. We prove that $τ_z$ uniquely extends t…
Starting from a finite-dimensional representation of the Yangian $Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's original presentation, we construct a Hopf algebra $X_\mathcal{I}(\mathfrak{g})$, 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 $X_\mathcal{I}(\mathfrak{g})\twoheadrightarrow Y(\mathfrak{g})$ whose kernel is generated as an ideal by the coefficients of a central matrix $\mathcal{Z}(u)$. When the underlying represe…
The statistical model of crystal melting represents BPS configurations of D-branes on a toric Calabi-Yau three-fold. Recently it has been noticed that an infinite-dimensional algebra, the quiver Yangian, acts consistently on the crystal-melting configurations. We physically derive the algebra and its action on the BPS states, starting with the effective supersymmetric quiver quantum mechanics on the D-brane worldvolume. This leads to remarkable combinatorial identities involving equivariant integrations on the moduli space of the quantum mechanics, which can be checked by numerical computation…
