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.

These talks will be on some topics in classical field theory and Riemannian geometry. The goal is to give a sketch of Kaluza-Klein theory, which is the idea that pure gravity on a spacetime endowed with extra dimensions, gives rise to Einstein gravity plus Yang-Mills theory (eg. electromagnetism) when reduced to usual spacetime. Initiated by Kaluza (1921) and Klein (1926), this theory still appears in various forms in modern theoretical physics.

Links: Slides for part 1

Suppose we are given a polygon in three-dimensional space whose sides have fixed length but whose joints are allowed to rotate freely. What can we say about the space of possible shapes of this polygon (i.e., the moduli space)? Although this seems like a simple question, the answer is somewhat surprising and combines ideas from algebraic geometry, symplectic geometry and several other areas of mathematics. It turns out that these spaces are examples of symplectic reductions which also have induced Kähler metrics. Furthermore, the techniques used to study the space of polygons can be applied to other moduli spaces.

A standard graded \(\mathbb{C}\)–algebra is said to be Koszul if the minimal graded free \(\mathbb{C}\)–resolution \(\textbf{F}\) is linear; by linear, we mean in every differential matrix of \(\textbf{F}\) every entry is a linear form. The definition of a Koszul algebra seems restrictive, but as it turns out, Koszul algebras possess remarkable numerical and homological properties and are ubiquitous in commutative algebra. Thus, it is an interesting question to determine families of rings that admit the Koszul property.

In this talk, we will briefly summarize some classical examples of Koszul algebras and discuss certain numerical invariants of the coordinate ring \(R\) of a generic collection of \(m\) lines in \(\mathbb{P}^n\). We also show that \(R\) is G-quadratic if \(2m \leq n+1\) and \(R\) is Koszul if \(m\) is even and \(m+1 \leq n\) or \(m\) is odd and \(m+2 \leq n\). Lastly, we show that if \[m > \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\] then \(R\) is not Koszul.

Hyperbolic lattices mean bilinear forms of signature \((n,1)\), the name reflecting the importance of hyperbolic space in their study. Reflective means that the isometry group is generated by reflections, up to finite index. The reflection groups arising this way are the hyperbolic version of finite and affine Weyl groups, and play a big role in Kac-Moody theory and the automorphism groups of algebraic surfaces. We have recently classified the reflective hyperbolic lattices of signature \((3,1)\), corresponding to hyperbolic $3$-space. I will explain this, and give an overview of the current state of the classification problem in higher dimensions.

Single- and multi-parameter persistence theory have been studied extensively and there are results from recent years analyzing the geometry of a data set via the appearance and disappearance of indecomposable parts in the algebraic structures associated to the data. So far, this has mainly been done with type \(\mathbb{A}\) quivers and their infinite counterparts (single-parameter) and quivers indexed over \(\mathbb{R}^2\) (multi-parameter).

In the case of tame- and wild-type quivers, it is difficult to write down and distinguish these indecomposable parts. In order to make these more complex quivers available for use in persistence theory, I aim to provide a framework for choosing particular subsets of indecomposables which may be distinguished from one another in a computationally feasible manner. I present the Kronecker quiver as a test case.

Links: Video recording

Even though complex polynomials are, in most ways, well known and well understood, there are still some surprising aspects that are only now being investigated. In this talk, I would like to highlight a new surprising connection between complex polynomials, the combinatorics of the noncrossing partitions and the combinatorial structures that Martin, Savitt and Singer call "basketballs". In particular, I will talk about a very nice combinatorially defined finite cell complex which compactifies the space of monic centered complex polynomials of degree d. This is joint work with Michael Dougherty. If there is time at the end of the talk, I will also explain the problem about braid groups that led Michael and I to undertake these investigations.

Links: Slides

How can we arrange lines to pass through the origin without creating sharp angles? This question arises in applications such as compressed sensing, wireless communication, and quantum information theory. For a fixed number of lines in a fixed dimension, an "optimal line packing" is one in which the sharpest angle is made as wide as possible. Examples include the 6 lines in \(\mathbb{R}^3\) that connect antipodes of a 20-sided die, and the 3 lines in \(\mathbb{R}^2\) that arise from the Mercedes-Benz logo. Many of the known optimal line packings feature remarkable symmetry. In particular, they arise as orbits of unitary representations of finite groups. In this talk, we investigate the structure of such symmetric line systems, and we demonstrate a method to recover the lines from their automorphism group. Our work leads to a partial classification of doubly transitive line packings.

Based on joint work with John Jasper and Dustin G. Mixon.

A *Kleinian* (or *du Val*, or *simple surface*) *singularity* \(X\) is the set of orbits \(\mathbb{C}^2/G\) where \(G\) is a finite group of 2x2 matrices of determinant \(1\). The set \(X\) is a two-dimensional affine variety whose ring of regular functions is \(\mathbb{C}[X]=\mathbb{C}[x,y]^G\), the ring of all \(G\)–invariant polynomials in two variables. Since the action of \(G\) on \(\mathbb{C}^2\) is not free (origin is fixed), \(X\) is a singular variety, hence the name. Kleinian singularities follow an ADE classification pattern.

Kleinian singularities lie at the crossroads of an overwhelming number of topics in geometry, algebra, and representation theory. As an example, the celebrated McKay correspondence gives a baffling connection between the sequence of blow-ups when resolving the singular point in \(X\), and tensor products of irreducible representations of \(G\).

In this talk we will focus on the *deformation quantization* \(\mathcal{O}^\lambda\) of \(\mathbb{C}[X]\) proposed by Crawley-Boevey and Holland in 1998. In the Type D case, for which Boddington provided an embedding \(\mathcal{O}^\lambda\) into a skew-group algebra. By adjusting this embedding, we obtain that \(\mathcal{O}^\lambda\) are examples of so called *principal Galois orders*, a large class of algebras that include the enveloping algebras of \(\mathfrak{gl}_n\). This allows us to classify all irreducible Harish-Chandra modules over \(\mathcal{O}^\lambda\). Furthermore we obtain a possibly new realization for \(\mathcal{O}^\lambda\) in Type D, in terms of Bernstein-Gelfand-Gelfand divided difference operators.

Simplicial objects are a fundamental tool in modern homotopy theory, (higher) category theory, and algebra. We will begin with a brief review / introduction, with an emphasis on the connection to categories, generalizing to the 2-Segal spaces of Dyckerhoff–Kapranov (also called decomposition spaces by Gálvez-Carillo–Kock–Tonks).

Any simplicial set has an underlying "outer face complex" obtained by forgetting about the degeneracy maps and the inner face maps. Our primary subject is the left adjoint to this forgetful functor, which freely adjoins inner faces and degeneracies to any outer face complex. A surprising fact is that this free functor takes any outer face complex to a decomposition space. Several previously-known examples of decomposition spaces, each of combinatorial origin and expressing a deconcatenation-type comultiplication, turn out to arise as such "free decomposition spaces." We identify the category of outer face complexes with a certain category of decomposition spaces over the monoid of natural numbers. This talk is based on joint work with Joachim Kock.

Links: arXiv:2210.11192

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