YANGIAN DEFORMATION OF THE LIE ALGEBRA dĩff(T2)

1998 ◽  
Vol 13 (20) ◽  
pp. 1611-1622
Author(s):  
Z. LIPIŃSKI
Keyword(s):  
Lie Algebra ◽  
Diffeomorphism Group ◽  
Baxter Equation ◽  
Lie Bialgebra ◽  
Quantum Double ◽  
R Matrix ◽  
Yangian Algebra

New solution of a classical Yang–Baxter equation on a loop extended, "modified" Lie algebra of a diffeomorphism group of a torus dĩff(T2)[λ]is given. Lie bialgebra structure on dĩff(T2)[λ] is defined. By quantization of a defined Lie bialgebra, a Yangian algebra Y(dĩff(T2)) is obtained. Existence of a quantum double and a quantum R-matrix for Y(dĩff(T2)) is considered.

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

scholarly journals A q-DEFORMATION OF VIRASORO AND U(1) KAC–MOODY ALGEBRAS WITH HOPF STRUCTURE

1999 ◽  
Vol 14 (12) ◽  
pp. 733-743 ◽  
Author(s):  
M. MANSOUR ◽  
E. H. TAHRI
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Baxter Equation ◽  
R Matrix ◽  
Unit Square

Using a general formalism of a q-deformation of an arbitrary Lie algebra, new kinds of q-deformed centerless Virasoro and U(1) Kac–Moody algebras are found. This q-deformation is associated to an R-matrix of unit square satisfying the quantum Yang–Baxter equation and allows a nontrivial Hopf structure. The central extension is also incorporated in this formalism.

scholarly journals Multiparametric and Colored Extensions of the Quantum Group GLq(N) and the Yangian Algebra Y(glN) Through a Symmetry Transformation of the Yang–Baxter Equation

1997 ◽  
Vol 12 (05) ◽  
pp. 945-962 ◽  
Author(s):  
B. Basu-Mallick ◽  
P. Ramadevi ◽  
R. Jagannathan
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Spectral Parameter ◽  
General Relation ◽  
Symmetry Transformation ◽  
Baxter Equation ◽  
Color Parameter ◽  
R Matrix ◽  
General Symmetry ◽  
Yangian Algebra

Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general "symmetry transformation" of the "particle conserving" R-matrix is found such that the resulting multiparametric R-matrix, with a spectral parameter as well as a color parameter, is also a solution of the Yang–Baxter equation (YBE). The corresponding transformation of the quantum YBE reveals a new relation between the associated quantized algebra and its multiparametric deformation. As applications of this general relation to some particular cases, multiparametric and colored extensions of the quantum group GL q(N) and the Yangian algebra Y(glN) are investigated and their explicit realizations are also discussed. Possible interesting physical applications of such extended Yangian algebras are indicated.

scholarly journals Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

2021 ◽  
Author(s):  
Raschid Abedin ◽  
Igor Burban
Keyword(s):  
Algebraic Geometry ◽  
Lie Algebra ◽  
Baxter Equation ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Infinite Dimensional ◽  
Geometric Context ◽  
Trigonometric Solution ◽  
Manin Triples ◽  
Geometric Study

AbstractThis paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.

scholarly journals Degenerate Sklyanin algebras

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Andrey Smirnov
Keyword(s):  
Difference Operators ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Rational Solutions ◽  
R Matrix ◽  
Gauge Transformations ◽  
Sklyanin Algebras ◽  
Sklyanin Algebra ◽  
The Difference

AbstractNew trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

Transfer Matrix of the Two-dimensional Ising Model

2020 ◽  
pp. 211-234
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Ising Model ◽  
Critical Point ◽  
Transfer Matrix ◽  
Functional Equations ◽  
Baxter Equation ◽  
Matrix Formalism ◽  
Two Dimensional ◽  
R Matrix ◽  
Transfer Matrix Formalism ◽  
Lowest Eigenvalue

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

Spectrum-generating bialgebra of the hexagonal-lattice dimer model

2009 ◽  
Vol 87 (10) ◽  
pp. 1099-1125 ◽  
Author(s):  
Jeffrey R. Schmidt
Keyword(s):  
Hexagonal Lattice ◽  
Baxter Equation ◽  
Dimer Model ◽  
R Matrix ◽  
The Matrix ◽  
Complete Set

The method of constructing a complete set of “zero-operator” identities preserved by the matrix coproduct is shown to be general, and is used to build the operator bialgebra for the hexagonal-lattice dimer model. This technique is complementary to the RTT-equation, and does not require a solution to the Yang–Baxter equation. The resulting bialgebra in the case of hexagonal lattice dimers has a distinctly Yangian structure, but has no R-matrix or antipode.

scholarly journals The EllipticGL(n)Dynamical Quantum Group as an𝔥-Hopf Algebroid

2009 ◽  
Vol 2009 ◽  
pp. 1-41 ◽  
Author(s):  
Jonas T. Hartwig
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Spectral Parameter ◽  
Exterior Algebra ◽  
Baxter Equation ◽  
Matrix Elements ◽  
Elliptic Solution ◽  
Hopf Algebroid ◽  
Quantum Dynamical ◽  
Hopf Algebroids

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

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