Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical matrices. A super dynamical matrix satisfies the zero weight condition if

    for all 

In this paper we classify super dynamical matrices with zero weight.

     

A class of solutions to the quantum colored Yang-Baxter equation

In this paper, we give a new method to solve the quantum colored Yang-Baxter matrix equation (QCYBE), and a class of solutions to the QCYBE is given.     

Retractability of set theoretic solutions of the Yang-Baxter equation

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-Z groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions.     

On Frobenius algebras and the quantum Yang-Baxter equation

It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya.

     

A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation

It is known that every skew-polynomial ring with generating set X and binomial relations in the sense of Gateva-Ivanova (Trans. Amer. Math. Soc. 343 (1994) 203) is an Artin-Schelter regular domain of global dimension |X|. Moreover, every such ring gives rise to a non-degenerate unitary set-theoretical solution of the quantum Yang-Baxter equation which fixes the diagonal of X². Gateva-Ivanova's conjecture (Talk at the International Algebra Conference, Miskolc, Hungary, 1996) states that conversely, every such solution R comes from a skew-polynomial ring with binomial relations. An equivalent conjecture (Duke Math. J. 100 (1999) 169) says that the underlying set X is R-decomposable. We prove these conjectures and construct an indecomposable solution R with |X|=∞ which shows that an extension to infinite X is false.     

GL3-invariant solutions of the Yang-Baxter equation and associated quantum systems

GL₃-invariant, finite-dimensional solutions of the Yang-Baxter equations acting in the tensor product of two irreducible representations of the group GL₃ are investigated. A number of relations are obtained for the transfer matrices which demonstrate the connection of representation theory and the Bethe Ansatz in GL₃-invariant models. Some of the most interesting quantum and classical integrable systems connected with GL₃-invariant solutions of the Yang-Baxter equation are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 92–121, 1982.     

Classification of constant solutions of the associative Yang-Baxter equation on Mat3

We find all nonequivalent constant solutions of the classical associative Yang-Baxter equation for Mat ₃ . New examples found in the classification yield the corresponding Poisson brackets on traces, double Poisson brackets on a free associative algebra with three generators, and anti-Frobenius associative algebras.     

Solutions of the Yang-Baxter equation

We give the basic definitions connected with the Yang-Baxter equation (factorization condition for a multiparticle S-matrix) and formulate the problem of classifying its solutions. We list the known methods of solution of the Y-B equation, and also various applications of this equation to the theory of completely integrable quantum and classical systems. A generalization of the Y-B equation to the case ofZ ₂-graduation is obtained, a possible connection with the theory of representations is noted. The supplement contains about 20 explicit solutions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 95, pp. 129–160, 1980.     

Spectral resolution of su(3)-invariant solutions of the Yang-Baxter equation

The spectral resolution of invariantR-matrices is computed on the basis of solution of the defining equation. Multiple representations in the Clebsch-Gordon series are considered by means of the classifying operator A: a linear combination of known operators of third and fourth degrees in the group generators. The matrix elements of A in a nonorthonormal basis are found. Explicit expressions are presented for the spectral resolutions for a number of representations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 145, pp. 3–21, 1985.     

On solutions to the Degasperis-Procesi equation

In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u₀(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively.     

Associative triples and the Yang-Baxter equation

We introduce triples of associative algebras as a tool for building solutions to the Yang-Baxter equation. It turns out that the class of R-matrices thus obtained is related to a Hecke-like condition, which is formulated in the framework of associative algebras with non-degenerate symmetric cyclic inner product. R-matrices for a subclass of theA _n-type Belavin-Drinfel’d triples are derived in this way.     

Classical limits of the SU(2)-invariant solutions of the Yang-Baxter equation

The possible classical limits of the SU(2)-invariant solution of the Yang-Baxter equation are systematically studied. In addition to the already known classical limits, namely the classical R-matrix, the lattice and the continuous L-operators, a series of new classical objects are introduced and the equations satisfied by them are enumerated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 119–136, 1985.     

The Cremmer-Gervais solution of the Yang-Baxter equation

A direct proof is given of the fact that the Cremmer-Gervais -matrix satisfies the (Quantum) Yang-Baxter equation

     

On the convergence of solutions of the enskog equation to solutions of the boltzmann equation

For the Enskog equation with a symmetrized kernel in a box an existence theorem is proved for initial data with finite mass, energy and entropy. Then by letting the diameter of the molecules go to zero we prove the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation.     

On some solutions to the Chapman-Kolmogorov integral equation

By applying integral transformations, we obtain some solutions to the Chapman-Kolmogorov equation. These are illustrated by examples.     

On periodic solutions to the von Foerster-Lasota equation

We show that the von Foerster-Lasota equation with parameter λ has a periodic solution in the space of Hölder continuous functions with exponent α if and only if α<λ. This generalizes the result from Dawidowicz and Haribash (Univ. Iagell. Acta Math. 37:321–324, 1999), in which existence of periodic solutions was proved only for λ>1.     

On the support of solutions to the Zakharov-Kuznetsov equation

In this article we prove that sufficiently smooth solutions of the Zakharov-Kuznetsov equation: