During Fall 2022, the seminar runs on Thursdays at 2:10pm–3:25pm in Carver 401. Grad students are especially encouraged to attend.

Topics include:

associative algebras and commutative rings,

representation theory and Lie theory,

connections to combinatorics, geometry and physics.

Talks

September 1 and 8, 2022

Twisting of function algebras of algebraic groups

Shlomo Gelaki (ISU)

Fix a complex linear algebraic group \(G\). Let \(\mathcal{O}(G)\) denote the function algebra of \(G\)
(it is a commutative Hopf algebra), and let \(\mathrm{Rep}(G)\) denote the representation category of \(G\)
(it is a symmetric tensor category). In my talk I will first explain why (ordinary) fiber
functors \(F:\mathrm{Rep}(G)\to\mathrm{Vect}\) correspond to Drinfeld twistings \(J\) of \(\mathcal{O}(G)\), namely to twisting Hopf
algebras \(\mathcal{O}(G)^J\), and then focus on the algebra structure and representation theory
of the (not necessarily commutative) Hopf algebras \(\mathcal{O}(G)^J\) and the one-sided twisted algebras
\(\mathcal{O}(G)_J\) for nilpotent \(G\). Finally, I will discuss some open problems and conjectures for arbitrary \(G\)
(e.g., solvable, reductive).

September 15, 2022

Moduli of Representations of Clannish Algebras

Cody Gilbert (University of Iowa)

We prove irreducible components of moduli spaces of semistable representations of clannish algebras are isomorphic to products of projective spaces. This is achieved by showing irreducible components of varieties of representations of clannish algebras can be viewed as irreducible components of skewed-gentle algebras, which we show are always normal. The main theorem generalizes an analogous result for moduli of representations of special biserial algebras proven by Carroll-Chindris-Kinser-Weyman.

We introduce the problem of controlling in small time a bilinear closed quantum system. We recall some known results when the dimension of the state space is finite [1], and give some new results concerning the infinite-dimensional case [2]. In particular, we remark the algebraic nature of this problem. We discuss these properties in several examples.