arXiv subjects

Travis Schedler

Publications and source records attributed to Travis Schedler.

61 records · Page 4Linked to original sources

arXiv2003

Trigonometric solutions of the associative Yang-Baxter equation

We classify trigonometric solutions to the associative Yang-Baxter equation (AYBE) for A = Mat_n, the associative algebra of n-by-n matrices. The AYBE was first presented in a 2000 article by Marcelo Aguiar and also independently by Alexandre Polishchuk. Trigonometric AYBE solutions limit to solutions of the classical Yang-Baxter equation. We find that such solutions of the AYBE are equal to special solutions of the quantum Yang-Baxter equation (QYBE) classified by Gerstenhaber, Giaquinto, and Schack (GGS), divided by a factor of q - q^{-1}, where q is the deformation parameter q = exp(h). In other words, when it exists, the associative lift of the classical r-matrix coincides with the quantum lift up to a factor. We give explicit conditions under which the associative lift exists, in terms of the combinatorial classification of classical r-matrices through Belavin-Drinfeld triples. The results of this paper illustrate nontrivial connections between the AYBE and both classical (Lie) and quantum bialgebras.

math.QA↗

arXiv2000

Proof of the GGS Conjecture

We prove the GGS conjecture (1993), due to Gerstenhaber, Giaquinto, and Schack, which gives a particularly simple explicit quantization of classical r-matrices for Lie algebras gl(n) in terms of an element R satisfying the quantum Yang-Baxter equation and the Hecke condition. The r-matrices were classified by Belavin and Drinfeld in the 1980s in terms of combinatorial objects known as Belavin-Drinfeld triples. We prove this conjecture by showing that the GGS matrix coincides with another quantization due to Etingof, Schiffmann, and the author, which is a more general construction. We do this by explicitly expanding the product from the aforementioned paper using detailed combinatorial analysis in terms of Belavin-Drinfeld triples.

math.QA↗

arXiv1999

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

We provide an explicit quantization of dynamical r-matrices for semisimple Lie algebras, classified earlier by the third author, which includes the Belavin-Drinfeld r-matrices. We do so by constructing an appropriate (dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group U_q(g), which twists the R-matrix of U_q(g) into the desired quatization. The construction of this twist is based on the method stemming from the work of Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on defining the twist as a unique solution of a suitable difference equation. This yields a simple closed formula for the twist. This construction allows one to confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture (about quantization of Belavin-Drinfeld r-matrices for sl(n) in the vector representation), which was stated earlier by the second author on the basis of computer evidence. It also allows one to define new quantum groups associated to semisimple Lie algebras. We expect them to have a rich structure and interesting representation theory.

math.QA↗

arXiv1999

On the GGS Conjecture

In the 1980's, Belavin and Drinfeld classified solutions r of the classical Yang-Baxter equation (CYBE) for simple Lie algebras \mathfrak g satisfying 0 \neq r + r_{21} \in (S^2 \mathfrak{g})^{\mathfrak{g}}. They proved that all such solutions fall into finitely many continuous families and introduced combinatorial objects to label these families, Belavin-Drinfeld triples. In 1993, Gerstenhaber, Giaquinto, and Schack attempted to quantize such solutions for Lie algebras \mathfrak{sl}(n). As a result, they formulated a conjecture stating that certain explicitly given elements R \in Mat_n(\mathbb C) \otimes Mat_n(\mathbb C) satisfy the quantum Yang-Baxter equation (QYBE) and the Hecke relation. Specifically, the conjecture assigns a family of such elements R to any Belavin-Drinfeld triple of type A_{n-1}. Following a suggestion from Gerstenhaber and Giaquinto, we propose an alternate form for R, given by R_J = q^{r^0} J^{-1} R_s J_{21} q^{r^0}, for a suitable twist J and a diagonal matrix r^0, where R_s is the standard Drinfeld-Jimbo solution of the QYBE. We formulate the ``twist conjecture'', which states that R_J = R_{\text{GGS}} and that R_J satisfies the QYBE. Since R_J by construction satisfies the Hecke relation, this conjecture implies the GGS conjecture. We check the twist conjecture by computer for n \leq 12 and show that it is true modulo \hbar^3. We provide combinatorial formulas for coefficients in the matrices R_J, R_{\text{GGS}} and prove both conjectures in the disjoint case---when Γ_1 \cap Γ_2 = \emptyset---and in the orthogonal generalized disjoint case, which is a generalization of Γ_1 \perp Γ_2. Finally, we prove the twist conjecture for the Cremmer-Gervais triple and discuss cases in which it is known that R_J = R_{\text{GGS}}.

math.QA↗

arXiv1999

Verification of the GGS conjecture for sl(n), n <= 12

In the 1980's, Belavin and Drinfeld classified non-unitary solutions of the classical Yang-Baxter equation (CYBE) for simple Lie algebras. They proved that all such solutions fall into finitely many continuous families and introduced combinatorial objects to label these families, Belavin-Drinfeld triples. In 1993, Gerstenhaber, Giaquinto, and Schack attempted to quantize such solutions for Lie algebras sl(n). As a result, they formulated a conjecture stating that certain explicitly given elements R satisfy the quantum Yang-Baxter equation (QYBE) and the Hecke condition. Specifically, the conjecture assigns a family of such elements R to any Belavin-Drinfeld triple of type A_{n-1}. Until recently, this conjecture has only been known to hold for n <= 4. In 1998 Giaquinto and Hodges checked the conjecture for n=5 by direct computation using Mathematica. Here we report a computation which allowed us to check that the conjecture holds for n <= 10. The program is included which prints an element R for any triple and checks that R satisfies the QYBE and Hecke conditions.

math.QA↗

arXiv1998

Set-theoretical solutions to the quantum Yang-Baxter equation

In 1992 V$.$Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions to the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set $X\times X$, where X is a fixed set. In this paper we study such solutions, which in addition satisfy the unitarity and nondegeneracy conditions. We discuss the geometric and algebraic interpretations of such solutions, introduce several constructions of them, and make first steps towards their classification.

math.QA↗

arXiv1997

On set-theoretical solutions of the quantum Yang-Baxter equation

Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the set $X\times X$, where $X$ is a fixed finite set. In this note we study such solutions, which satisfy the unitarity and the crossing symmetry conditions -- natural conditions arising in physical applications. More specifically, we consider ``linear'' solutions: the set $X$ is an abelian group, and the map $R$ is an automorphism of $X\times X$. We show that in this case, solutions are in 1-1 correspondence with pairs $a,b\in \End X$, such that $b$ is invertible and $bab^{-1}=\frac{a}{a+1}$. Later we consider ``affine'' solutions ($R$ is an automorphism of $X\times X$ as a principal homogeneous space), and show that they have a similar classification. The fact that these classifications are so nice leads us to think that there should be some interesting structure hidden behind this problem.

q-alg↗