On the R-matrix realization of Yangians and their representations

Abstract

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) which implies the corresponding theorem of Drinfeld for the Yangians Y(a). We also give explicit constructions for all fundamental representation of the Yangians.

65 pages

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• Parabolic presentations of the Yangian Y(gl_n)
• General boundary conditions for the sl(N) and sl(M|N) open spin chains
• Yangians, Integrable Quantum Systems and Dorey's rule
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• Yangian of the Queer Lie Superalgebra
• R-matrix presentation for (super)-Yangians Y(g)
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Cited by in corpus (41)

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• Equivalences between three presentations of orthogonal and symplectic Yangians
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Yangian symmetry of the Y=0 maximal giant graviton
• Twisted Yangians for symmetric pairs of types B, C, D
• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• Representations of twisted Yangians of types B, C, D: I
• On the reflection of magnon bound states
• Bethe vectors for orthogonal integrable models
• Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
• Spinorial R operator and Algebraic Bethe Ansatz
• Representations of the super Yangians of types $A$ and $C$
• Yangians and Yang-Baxter R-operators for ortho-symplectic superalgebras
• Vertex Representations for Yangians of Kac-Moody algebras
• Yangian characters and classical W-algebras
• $h$-adic quantum vertex algebras associated with rational $R$-matrix in types $B$, $C$ and $D$
• Representations of twisted Yangians of types B, C, D: II
• Twisted Yangians of small rank
• Representations of the Yangians associated with Lie superalgebras $\mathfrak{osp}(1|2n)$
• Nested algebraic Bethe ansatz for orthogonal and symplectic open spin chains
• Representations of the orthosymplectic Yangian
• A Drinfeld-type presentation of the orthosymplectic Yangians
• Yangian doubles of classical types and their vertex representations
• Superspin Chains Solutions from 4D Chern-Simons Theory
• Isomorphism between two realizations of the Yangian Y(so_3)
• Manin Matrices for Quadratic Algebras
• Principal Realization of the Yangian Y(gl(n))
• Eigenvalues of Bethe vectors in the Gaudin model
• From Twisted Quantum Loop Algebras to Twisted Yangians
• Orthogonal and symplectic Yangians - linear and quadratic evaluations
• Representations of the super-Yangian of type $B(n,m)$
• Scalar products and norm of Bethe vectors in ${\mathfrak{o}}_{2n+1}$ invariant integrable models
• Representations of Quantum Affine Algebras in their $R$-Matrix Realization
• Representations of the super-Yangian of type $D(n,m)$
• Orthosymplectic Yangians
• Gaussian generators for the Yangian associated with the Lie superalgebra $\mathfrak{osp}(1|2m)$
• Quantum Representation Theory and Manin matrices I: finite-dimensional case
• The R-matrix formalism for quantized enveloping algebras