Let $G_M$ be one of the complex Lie groups $O_M$ and $Sp_M$. The irreducible finite-dimensional representations of the group $G_M$ are labeled by partitions $μ$ satisfying certain extra conditions. Let $U$ be the representation of $G_M$ corresponding to $μ$. Regard the direct product $G_N\times G_M$ as a subgroup of $G_{N+M}$. Let $V$ be the irreducible representation of $G_{N+M}$ corresponding to a partition $λ$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $G_N$. Let $n$ be sum of parts of $λ$ less the sum of parts of $μ$. For any choice of a standard Young tableau of skew shape $λ/μ$, we realize $W$ as a subspace in the tensor product of $n$ copies of the defining $N$-dimensional representation of $G_N$. This subspace is determined as the image of a certain linear operator $F(M)$ in the tensor product, given by an explicit formula. When M=0 and $W=V$ is an irreducible representation of $G_N$, we recover the classical realization of $V$ as a subspace in the space of all traceless tensors. Then the operator F(0) can be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the chosen standard tableau of shape $λ$. Even in the special case M=0, our formula for the operator $F(M)$ is new. Our results are applications of representation theory of the twisted Yangian, corresponding to $G_N$. In particular, $F(M)$ is an intertwining operator between two representations of the twisted Yangian in the $n$-fold tensor product.
The Yangian of the Lie algebra $gl_N$ has a distinguished family of irreducible finite-dimensional representations, called elementary representations. They are parametrized by pairs, consisting of a skew Young diagram and a complex number. Each of these representations has an explicit realization, it extends the classical realization of the irreducible polynomial representations of $gl_N$ by means of the Young symmetrizers. We explicitly construct analogues of these elementary representations for the twisted Yangian, which corresponds to the Lie algebra $so_N$. Our construction provides soluti…
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 initiate a theory of highest weight representations for twisted Yangians of types B, C, D and we classify the finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types CI, DIII and BCD0.
We study the tensor product $W$ of any number of "elementary" irreducible modules $V_1,...,V_k$ over the Yangian of the general linear Lie algebra. Each of these modules is determined by a skew Young diagram and a complex parameter. For any indices $i,j=1,...,k$ there is a canonical non-zero intertwining operator $A_{ij}$ between the tensor products $V_i\otimes V_j$ and $V_j\otimes V_i$. This operator is defined up to a scalar multipler. We show that the tensor product $W$ is irreducible, if and only if all operators $A_{ij}$ with $i<j$ are invertible. This implies that the Yangian module $W$ …
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 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.
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.
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 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 give the principal realization of the twisted Yangians of orthogonal and symplectic types. The new bases are interpreted in terms of discrete Fourier transform over the cyclic group Z_N.
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 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.
Sapienza Università di Roma（罗马大学）；University of Zagreb Faculty of Science, Department of Mathematics（萨格勒布大学理学院）；Central China Normal University（华中师范大学）
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.
Let $GL_M$ be general linear Lie group over the complex field. The irreducible rational representations of the group $GL_M$ are labeled by pairs of partitions $μ$ and $\tildeμ$ such that the total number of non-zero parts of $μ$ and $\tildeμ$ does not exceed $M$. Let $U$ be the representation of $GL_M$ corresponding to such a pair. Regard the direct product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the irreducible rational representation of the group $GL_{N+M}$ corresponding to a pair of partitions $λ$ and $\tildeλ$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a…
This is a review paper on the algebraic structure and representations of the A type Yangian and the B, C, D types twisted Yangians. Some applications to constructions of Casimir elements and characteristic identities for the corresponding Lie algebras are also discussed.
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.
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。
Let g be a complex simple Lie algebra and let V be a finite dimensional U(g) module. A relative Yangian is defined with respect to this pair. According to recent work of Khoroshkin and Nazarov the finite dimensional simple modules of the Yangians for g = sl(n) or the twisted Yangians for g = sp(2n); so(n) are described by the simple modules of relative Yangians for some V using the appropriate simple Lie algebra g. Here a classification of the simple modules of a relative Yangian is obtained simply and briefly as an advanced exercise in Frobenius reciprocity inspired by a Bernstein- Gelfand eq…
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 prove an analogue of the Sylvester theorem for the generator matrices of the quantum affine algebra ${\rm U}_q(\hat{\mathfrak{gl}}_n)$. We then use it to give an explicit realization of the skew representations of the quantum affine algebra. This allows one to identify them in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (or $q$-character). We also apply the quantum Sylvester theorem to construct a $q$-analogue of the Olshanski algebra as a projective limit of certain centralizers in ${\rm U}_q(\mathfrak{gl}_n)$ and show that this …
A major open problem in the theory of twisted commutative algebras (tca's) is proving noetherianity of finitely generated tca's. For bounded tca's this is easy, in the unbounded case, noetherianity is only known for Sym(Sym^2(C^\infty)) and Sym(\wedge^2(C^\infty)). In this paper, we establish noetherianity for the skew-commutative versions of these two algebras, namely \wedge(Sym^2(C^\infty)) and \wedge(\wedge^2(C^\infty)). The result depends on work of Serganova on the representation theory of the infinite periplectic Lie superalgebra, and has found application in the work of Miller-Wilson on…
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
For an arbitrary split adjoint group we identify the unramified Whittaker space with the space of skew-invariant functions on the lattice of coweights and deduce from it the Casselman-Shalika formula.
We study the eigenproblem for Bethe subalgebras of the Yangian $Y(\mathfrak{gl}_n)$ in tame representations, i.e. in finite dimensional representations which admit Gelfand-Tsetlin bases. Namely, we prove that for any tensor product of skew modules $V=\otimes_{i=1}^k V_{λ_i \setminus μ_i}(z_i)$ over the Yangian $Y(\mathfrak{gl}_n)$ with generic $z_i$'s, the family of Bethe subalgebras $B(X)$ with $X$ being a regular element of the maximal torus of $GL_n$ (or, more generally, with $X \in \overline{M_{0,n+2}}$) acts with a cyclic vector on $V$. Moreover, for $X$ in the real form of $\overline{M_{…
We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted chain maps allow us to transfer information between resolutions for use in homology theories, for example, in those governing deformation theory. We show how to translate in particular from the default (but often cumbersome) reduced bar resolution to a more convenient twisted product resolution. This provides a more universal approach to so…
The deformation theory of an algebra is controlled by the Gerstenhaber bracket, a Lie bracket on Hochschild cohomology. We develop techniques for evaluating Gerstenhaber brackets of semidirect product algebras recording actions of finite groups over fields of positive characteristic. The Hochschild cohomology and Gerstenhaber bracket of these skew group algebras can be complicated when the characteristic of the underlying field divides the group order. We show how to investigate Gerstenhaber brackets using twisted product resolutions, which are often smaller and more convenient than the cumber…
