College of Liberal Arts & Sciences

# Algebra Seminar

**Abstract: **

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is of utmost importance when studying representations of $\mathfrak{g}$. In 2010, V. Futorny and S. Ovsienko gave a realization of $U(\mathfrak{gl}_n)$ as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of $S_1\times S_2\times\cdots\times S_n$ where $S_j$ is the symmetric group on $j$ variables. With some connections to Galois Theory, an interesting question is what would a similar object be in the invariant ring with respect to a product of Alternating groups? We will discuss such an object and some results about its representations and how they relate to $U(\mathfrak{gl}_n)$ representations. This is based on the work in my recent preprint arXiv:1907.13254