Representations Of Yangian Research Articles

Poles of finite-dimensional representations of Yangians

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of $V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.

Finite-Dimensional Representations of Yangians in Complex Rank

Abstract We classify the “finite-dimensional” irreducible representations of the Yangians $Y(\mathfrak{g}\mathfrak{l}_t)$ and $Y(\mathfrak{s}\mathfrak{l}_t)$. These are associative ind-algebras in the Deligne category $\textrm{Rep}(GL_t)$, which generalize the regular Yangians $Y(\mathfrak{g}\mathfrak{l}_n)$ and $Y(\mathfrak{s}\mathfrak{l}_n)$ to complex rank. They were first defined in the paper [14]. Here we solve [14, Problem 7.2]. We work with the Deligne category $\textrm{Rep}(GL_t)$ using the ultraproduct approach introduced in [7] and [16].

Rational and polynomial representations of Yangians

We define natural classes of rational and polynomial representations of the Yangian of the general linear Lie algebra. We also present the classification and explicit realizations of all irreducible rational representations of the Yangian.

On representations of Yangian of Lie Superalgebra A(n,n) type

The finite-dimensional irreducible representations of Yangian of Lie Superalgebra of A(n, n) type is described in terms of Drinfel'd polynomials. The necessarily and sufficient conditions of finite-dimensionality of irreducible representation are formulated and proved. The Poincare-Birkoff-Witt theorem for Yangian of A(n, n) Lie Superalgebra is proved also.

Mickelsson algebras and representations of Yangians

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.

Irreducible representations of 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.

Baxter Q-operators and representations of Yangians

We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang–Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang–Baxter equation serve as elementary, “partonic” building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider gl ( n ) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe Ansatz techniques.

Many-spinon states and representations of Yangians in the SU(n) Haldane–Shastry model

We study the relation of Yangians and spinons in the SU(n) Haldane–Shastry model. The representation theory of the Yangian is shown to be intimately related to the fractional statistics of the spinons. We construct the spinon Hilbert space from tensor products of the fundamental representations of the Yangian.

Quantum, higher-spin, local charges in symmetric space sigma models

Potential anomalies are analysed for the local spin-3 and spin-4 classically conserved currents in any two-dimensional sigma model on a compact symmetric space $G/H$, with $G$ and $H$ classical groups. Quantum local conserved charges are shown to exist in exactly those models which also possess quantum non-local (Yangian) charges. The possibility of larger sets of quantum local charges is discussed and shown to be consistent with known S-matrix results and the behaviour of the corresponding Yangian representations.

Skew Schur Functions and Yangian Actions on Irreducible Integrable Modules of $ \hat{\frak g \frak l} _N $

An action of the Yangian of the general linear Lie algebra \( {\frak g \frak l} _N \) is defined on every irreducible integrable highest weight module of \( \hat{\frak g \frak l} _N \). This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra \( \frak H \). Each vacuum module of \( \hat{\frak g \frak l} _N \) is decomposed into irreducible Yangian representations by means of the intertwiners of \( \frak H \). Components of this decomposition are parameterized by semi-infinite skew Young diagrams. The decomposition gives rise to a character formulas for the modules of \( \hat{\frak g \frak l} _N \) in terms of skew Schur functions.

Drinfeld Functor and Finite-Dimensional Representations of Yangian

We extend the results of Drinfeld on Drinfeld functor to the case l>n. We present the character of finite-dimensional representations of the Yangian Y(sl_n) in terms of the Kazhdan-Lusztig polynomials as a consequence.

Representations of Yangians with GelfandZetlin bases

We study certain family of finite-dimensional modules over the Yangian $Y(gl_N)$. The algebra $Y(gl_N)$ comes equipped with a distinguished maximal commutative subalgebra $A(gl_n)$ generated by the centres of all algebras in the chain $Y(gl_1)\subset Y(gl_2)\subset...\subset Y(gl_N)$. We study the finite-dimensional $Y(gl_N)$-modules with a semisimple action of the subalgebra $A(gl_N)$. We call these modules tame. We provide a characterization of irreducible tame modules in terms of their Drinfeld polynomials. We prove that every irreducible tame module splits into a tensor product of modules corresponding to the skew Young diagrams and some one-dimensional module. The eigenbases of $A(gl_N)$ in irreducible tame modules are called Gelfand-Zetlin bases. We provide explicit formulas for the action of the Drinfeld generators of the algebra $Y(gl_N)$ on the vectors of Gelfand-Zetlin bases.

YANGIANS AND FINITE W-ALGEBRAS

Using an algebra morphism between Yangians and some finite W-algebras, we determine the (product of) finite dimensional representations of the latters. This demonstrates a correspondence between evaluation representations of Yangians and Miura representations of W-algebras.

Combinatorial Structure of Finite Dimensional Representations of Yangians: the Simply-Laced Case

We compute the decomposition of representations of Yangians into g-modules for simply-laced g. The decomposition has an interesting combinatorial tree structure. Results depend on a conjecture of Kirillov and Reshetikhin.

The [formula omitted] WZW models Spinon decomposition and yangian structure

We present a ‘spinon formulation’ of the SU(n) 1 Wess-Zumino-Witten models. Central to this approach are a set of massless quasi-particles, called ‘spinons’, which transform in the representation n of su( n) and carry fractional statistics of angle θ = π n . Multi-spinon states are grouped into irreducible representations of the yangian Y( sl n ). We give explicit results for the su( n) content of these yangian representations and present N-spinon cuts of the WZW character formulas. As a by-product, we obtain closed expressions for characters of the su( n) Haldane-Shastry spin chains.

Transfer Matrix and Yangian Related to Chaplygin-Goryachev Top

A new representation of Yangian for sl(2) is obtained from e(3) algebra. For the given rational solution of the quantum Yang-Baxter equation the algebraic relations of the transfer matrix is discussed and a new transfer matrix related to Chaplygin-Goryachev top are constructed in terms of the generators of Yangian.

Yangians, integrable quantum systems and Dorey's rule

We study tensor products of fundamental representations of Yangians and show that the fundamental quotients of such tensor products are given by Dorey's rule.

Spectra of Tensor Products of Finite Dimensional Representations of Yangians

Spectra of Tensor Products of Finite Dimensional Representations of Yangians

Finite-difference representations of the degenerate affine Hecke algebra

The representations of the degenerate affine Hecke algebra in which the analogues of the Dunkl operators are given by finite-difference operators are introduced. The non-self-adjoint lattice analogues of the spin Calogero-Sutherland Hamiltonians are analysed by a Bethe ansatz. The gl( m) Yangian representations arising from the finite-difference representations of the degenerate affine Hecke algebra are shown to be related to the Yangian representation of the 1-d Hubbard model.

Gelfand-Tsetlin basis for representations of Yangians

We construct an analog of the Gelfand-Tsetlin basis in the finite-dimensional irreducible representations of Yangians and find the matrix elements of the Drinfeld generators in this basis. As a special case of this construction, we obtain the well-known Gelfand-Tsetlin basis in the representations of the Lie algebra gl(N).