Title：Perturbative expansion of Yang-Baxter operators and Lie algebra cohomology

Reporter：Emanuele Zappala

Work Unit：Idaho State University

Time：2023/10/19 20:00-22:00

Address：Zoom id：904 645 6677 Password:2023

**Summary of the report: **

Self-distributive objects in symmetric monoidal categories are known to produce Yang-Baxter operators, e.g. quandles and their linearization in modules. An important class of self-distributive objects arises from Lie algebras (binary and $n$-ary as well), giving a way of constructing a Yang-Baxter operator from an initial Lie algebra. This allows us to study the relation between self-distributive cohomology, Lie algebra cohomology, and Yang-Baxter cohomology. We therefore obtain a theory of infinitesimal deformations of Yang-Baxter operators that is tightly connected to the deformation theory of Lie algebras and their associated self-distributive objects. In this talk, I will explore these connections and the results that can be shown in this regard. Moreover, I will consider the problem of obtained higher order deformations of the Yang-Baxter operators in relation to the higher deformations of Lie algebras, therefore producing Yang-Baxter operators that are given by a formal power series (perturbative expansion). I will also present several open questions.

**Introduction of the Reporter: **

Emanuele Zappala is an assistant professor in the department of Mathematics and Statistics at Idaho State University. His work mainly concerns quantum algebra (Yang-Baxter operators and their cohomology), geometric topology (cohomological invariants of embedded surfaces), applications of algebra and topology to quantum machine learning, and operator learning (iterative methods in deep learning).