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Register a new userA Unified Algebraic Approach to Classical Yang-Baxter Equation
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Authors
Chengming Bai
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In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parakahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry as a Lie bracket ``left-twisted by a classical r-matrix.
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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 parakahler 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 parakahler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called ${cal O}$-operator, whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an ${cal O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parakahler Lie algebras) from ${cal O}$-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, parakahler Lie groups correspond to Poisson-Lie groups in this sense.
We construct the 1D $mathbb{Z}_3$ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the $mathbb{Z}_3$ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the $mathbb{Z}_3$ parafermionic model is a direct generalization of 1D $mathbb{Z}_2$ Kitaev model. Both the $mathbb{Z}_2$ and $mathbb{Z}_3$ model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian $hat{H}_{123}$ based on Yang-Baxter equation. Different from the Majorana doubling, the $hat{H}_{123}$ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, $omega$-parity $P$($omega=e^{{textrm{i}frac{2pi}{3}}}$) and emergent parafermionic operator $Gamma$, which are the generalizations of parity $P_{M}$ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the $mathbb{Z}_3$ parafermionic model and $hat{H}_{123}$ can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.
A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.
We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfeld classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly construct for all n > 2 a boundary solution based on the maximal parabolic subalgebra of sl(n) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to $n$. We also give examples of non-boundary solutions for the classical simple Lie algebras.
In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the Yang-Baxter equation. Such solutions are degenerate in general, and thus different from solutions obtained from braces and other algebraic structures. Our main result concerns a description of a set-theoretic solution of the Yang-Baxter equation, obtained from an arbitrary skew lattice. We also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets.
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