Bethe subalgebras of the Yangian Y(gl(n)), Gelfand-Tsetlin patterns, and the cactus group

One prevailing technique in the representation theory of quantum groups is using maximal commutative subalgebras to decompose representations. We will look at one such family, the Bethe subalgebras of the Yangian of gl(n), and describe their parameter space as well as how they can be used to decompose tame representations of Y(gl(n)). Each Bethe subalgebra corresponding to a real parameter acts with simple spectrum, and the resulting eigenlines can be identified with combinatorial Gelfand-Tsetlin keystone patterns, which carry a gl(n) crystal structure. This is joint work with Anfisa Gurenkova and Lenya Rybnikov.