Yangians of classical Lie superalgebras: basic constructions, quantum double and universal R-matrix (original) (raw)

The Yangian of the strange Lie superalgebra and its quantum double

Theoretical and Mathematical Physics, 2013

We construct the Yangian of the strange Lie superalgebra as a particular case of the general construction of a twisted Yangian. We describe a Poincaré-Birkhoff-Witt basis of the Yangian of the type-Qn Lie superalgebra and construct the quantum double of the Yangian of the type-Q2 strange Lie superalgebra.

On representations of Yangian of Lie SuperalgebraA(n,n) type

Journal of Physics: Conference Series, 2013

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. 0 This work is partially supported by Federal Program "Scientific and scientific-educational personnel of innovative Russia event 1.2.2 (contract number P116) and by Federal Program "Scientific and scientific-educational personnel of innovative Russia"(agreement № 14.A18.

The Yangian of and its quantum R-matrices

J High Energy Phys, 2011

In this paper we study Yangians of sl(n|m) superalgebras. We derive the universal R-matrix and evaluate it on the fundamental representation obtaining the standard Yang R-matrix with unitary dressing factors. For m = 0, we directly recover up to a CDD factor the well-known S-matrices for relativistic integrable models with su(n) symmetry. Hence, the universal R-matrix found provides an abstract plug-in formula, which leads to results obeying fundamental physical constraints: crossing symmetry, unitrarity and the Yang-Baxter equation. This implies that the Yangian double unifies all desired symmetries into one algebraic structure. In particular, our analysis is valid in the case of sl(n|n), where one has to extend the algebra by an additional generator leading to the algebra gl(n|n). We find two-parameter families of scalar factors in this case and provide a detailed study for gl(1|1).

The gl (M|N) super Yangian and its finite-dimensional representations

Letters in Mathematical Physics, 1996

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.

Representations of super Yangian

Journal of Mathematical Physics, 1995

The classification of the finite-dimensional irreducible representations of the super Yangian associated with the Lie superalgebra gl(1‖1) is presented in detail.

On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

Communications in Mathematical Physics, 2008

We study classical twists of Lie bialgebra structures on the polynomial current algebra g [u], where g is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of g. We give complete classification of quasi-trigonometric r-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of sl(n).

Drinfeld Yangian of the queer Lie superalgebra sq 1

Journal of Physics: Conference Series, 2019

Drinfeld Yangian of a queer Lie superalgebra sq1 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 case Lie superalgebra sq1 that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev-Reshetikhin-Takhtadzhjan approach, are isomorphic.