In this talk we investigate how to get frieze patterns from the $A_\infty$-curve singularity, via the category of graded maximal Cohen-Macaulay modules $CM^{\mathbb{Z}}(\mathbb{C}[x,y]/(x^2))$. This is a Frobenius category and was studied in the context of categories for Grassmannian cluster algebras and triangulations of the infinity-gon by August, Cheung, Faber, Gratz, and Schroll. Extending the cluster character from work of Paquette and Yildirim to this setting we obtain a new type of infinite friezes that can be related to Penrose tilings. This is joint work in progress with Özgür Esentepe.

Let G be a linear algebraic group acting on a vector space V. We consider the problem to compute the stabilizer in G of a given element of V, or, more generally, of a subspace of V. The ground field is assumed to be algebraically closed of characteristic 0. Then the problem reduces to computing the component group of the stabilizer, that is, to determine one element of G lying in each component of the stabilizer. We have developed computational techniques, implemented in the computational algebra system GAP4. We will illustrate them for computation of the stabilizer of a nilpotent element in a simple complex Lie algebra. This is joint work with Emanuele Di Bella.

Let $p$ be a prime number and $K$ be a field containing a root of $1$ of order $p^n$ for every $n\in \mathbb{N}$. Based on a previous conjecture by F. Bogomolov, L. Positselski conjectured in 2005, that the commutator subgroup of the maximal pro-$p$ Galois group $G_K(p)$ is a free pro-$p$ group. This subgroup corresponds to to the field $\sqrt[p^\infty]{K}:=K(\sqrt[p^n]{a}:n\in \mathbb{N},a\in K)$. There are large classes of fields, where this conjecture is known to hold, e.g., local and global fields and fields whose maximal pro-$p$ Galois group is of elementary type, but the general case is still open. Positselski showed that the conjecture would follow from strong Koszulity properties of the Milnor K-theory of $K$ mod $p$. In 2022 C. Quadrelli and T. Weigel gave another criterion for the afore mentioned conjecture depending (in a sophisticated way) on only two Galois cohomology groups, which allowed them to prove the conjecture for groups of elementary type. In their paper they also asked if there exists a connection between these two seemingly unrelated approaches. In this talk we would like to present a theorem linking the two results and some consequences of it. If time permits we also present a proof of Positselski's Module Koszulity Conjecture 1 for pro-$p$ groups of elementary type.

Let $\mathfrak{g}$ be a Lie algebra and $\ell \in \mathfrak{g}^\ast$ be a linear form on $\mathfrak{g}$. The radical of $\ell$ is given by $\mathfrak{g}(\ell)=\{ x \in \mathfrak{g} \mid \ell[y,x]=0\mbox{ for all }y \in \mathfrak{g} \}$ and is a subalgebra of $\mathfrak{g}$. The index of $\mathfrak{g}$ is then defined as $ind(\mathfrak{g}) = \inf \{ \dim \mathfrak{g}(\ell) \mid \ell \in \mathfrak{g}^\ast \}$. After having presented some of the known result about the index of Lie algebras, we will focus on nilpotent Lie algebras and show how one can determine the index for several families of nilpotent Lie algebras, such as the class of filiform Lie algebras and certain families of free nilpotent Lie algebras. This is joint work with Dietrich Burde (University of Vienna).

In this joint work with Friedrich Wagemann we will compute the adjoint cohomology spaces $H^n(L,L)$ for the family of perfect, but not semisimple Lie algebras $L=sl_2(C)\ltimes V_n$, where $V_n$ denotes the irreducible representation of $\mathfrak{sl}_2(\mathbb C)$ of dimension $n+1$. The cohomology of such Lie algebras is of interest in the context of a conjecture by Pirashvili, which says that a perfect Lie algebra is semisimple if and only if all the adjoint cohomology vanishes. The computation involves the Hochschild-Serre spectral sequence and the explicit plethysm formula for the multiplicities of the irreducible representations $V_{jk-2n}$ arising as a direct summand in the decompostion of exterior product $\Lambda^j(V_{j+k-1})$ into irreducible modules. The multiplicities can be expressed in terms of the partition numbers $p(j,k,n)$, the number of partitions of $n$ into at most $k$ parts, having largest part at most $j$. We will prove certain partition identities for this purpose.

Koszul algebras form a mysterious class of graded algebras defined by quadratic relations, appearing in various areas of mathematics. Motivated by questions in Galois theory, we explore the properties of Koszul Lie algebras, with a particular focus on their subalgebras. To this end, we employ HNN-extensions as a key tool to decompose such Lie algebras.