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- Title
Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules.
- Authors
Cañadas, Agustín Moreno; Fernández Espinosa, Pedro Fernando; Ballester-Bolinches, Adolfo
- Abstract
The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form A x = B x ′ , where A and B are m × n , F -matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions.
- Subjects
YANG-Baxter equation; NONASSOCIATIVE algebras; DIFFERENTIAL forms; QUANTUM computing; KNOT theory; CONTINUED fractions; INDECOMPOSABLE modules; QUANTUM groups
- Publication
Computation, 2023, Vol 11, Issue 3, p43
- ISSN
2079-3197
- Publication type
Article
- DOI
10.3390/computation11030043