Welcome to the Algebra and Geometry Seminar at Iowa State University, organized by Jonas Hartwig, Jason McCullough, and Tathagata Basak.

During Spring 2023, the seminar runs on **Thursdays at 2:10pm–3:00pm** (plus optional discussion) 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.

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).

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.

Links: arXiv paper and lecture notes

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.

[1] D'Alessandro: Small time controllability of systems on compact Lie groups and spin angular momentum, J. Math. Phys. 42, 4488 (2001)

[2] Chambrion and Pozzoli: Small-time bilinear control of Schrödinger equations with application to rotating linear molecules, preprint (arXiv: 2207.03818)

We introduce to the notion of vertex operator algebras \(V\) and a few important categories of \(V\)–modules, in particular the so called logarithmic modules \(M\) which are important for logarithmic conformal field theory. We give an explicit construction of how these vertex operators act on \(M\) for the triplet vertex operator algebra \(W(p)\) and also give a construction of all irreducible modules for certain higher rank cases for \(V\) given by the kernel of the intersection of multiple screening operators that arise from higher rank lattice vertex operator algebras.

Consider the complex groups \(G=GL(n,\mathbb{C}),\, SO(n,\mathbb{C})\). We can embed \(G_{n-1}:=GL(n-1,\mathbb{C}),\, SO(n-1,\mathbb{C})\) into \(G\) in a natural way. Let \(B_{n-1}\) be a Borel subgroup of \(G_{n-1}\subset G\). Then it follows from the theory of spherical pairs that \(B_{n-1}\) acts on the flag variety \(\mathcal{B}\) of \(G\) with finitely many orbits. In this talk, we discuss the geometric and combinatorial properties of these orbits and how they can be applied to study Gelfand-Zeitlin modules. In particular, for \(G=GL(n,\mathbb{C})\) we develop a bijection between the set of \(B_{n-1}\)–orbits on \(\mathcal{B}\) and partitions of the set \(\{1,\dots, n\}\) into ordered subsets. This bijection allows us to obtain an explicit formula for the number of \(B_{n-1}\)–orbits on \(\mathcal{B}\) using the classical Lah numbers and find explicit representatives for the orbits in terms of flags. We also discuss a similar picture for the orthogonal group. Using these combinatorial models and work of Richardson and Springer, we can understand the closure relations for \(B_{n-1}\)–orbits on \(\mathcal{B}\). This is a key step in geometrically constructing a category of modules related to Gelfand-Zeitlin modules.

Field theory is a branch of physics in which all the fundamental interactions of nature can be formulated. In the first talk I hope to explain a few of the key concepts from classical field theory including Lagrangians, Euler-Lagrange equation, symmetry and Noether’s theorem, gauge fields, using mainly the example of electromagnetism, starting with the Dirac equation for the electron field. In the second talk I will say something about the corresponding mathematical language of principal fiber bundles with a connection, and generalizations to noncommutative geometry.

In the 24-dimensional Euclidean space, there are 24 lattices such that the squared length of every lattice vector is an even integer and such that the "fundamental parallelogram" formed by a basis of the lattice has volume 1. I will try to explain the reason for this using some highly symmetric analytic functions on the upper half plane, called modular forms. We will end with a puzzle about these 24 lattices that does not have a good explanation.

The quantum group associated with a complex Lie algebra is a Hopf algebra for which the coradical (i.e., the biggest cosemisimple part) is an abelian group algebra. From that point of view, the quantum group is a pointed Hopf algebra over an abelian group. In the early 2000s, Andruskiewitsch-Schneider designed a strategy to obtain a classification of pointed Hopf algebras by studying two invariants: the coradical and certain braided subalgebra. This strategy led to a complete classification of finite-dimensional pointed Hopf algebras with abelian coradical.

We will begin this talk with a brief review of the main ingredients that led to the classification in the abelian group case. The braided objects aforementioned are generalizations of the positive parts of small quantum groups, known as Nichols algebras. As happens for quantum groups, we will see that these Nichols algebras are equipped with several combinatorial structures, which are fundamental for the classification.

Then we will focus on the classification problem in the non-abelian group realm, where the main obstruction is the lack of a complete understanding of the Nichols algebras. However, a big family of finite-dimensional Nichols algebras over non-abelian groups were recently classified by Heckenberger-Vendramin, again using combinatorial structures reminiscent of those available for quantum groups. We will report on joint work with Angiono and Lentner, where we classified all Hopf algebras over these Nichols algebras. We will explain how we could translate several results known for the abelian case to our setting using the folding construction for Nichols algebras due to Lentner, which relates to equivariantization/de-equivariantization for Hopf algebras.

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