I will give an introduction to the higher-order hom-associative Weyl algebras and show how a conjecture about them is related to the still open Generalized Dixmier Conjecture. (pdf)

Hilberts basis theorem is a classic result that says R[x] is Noetherian if R is Noetherian, where R is an associative ring. The theorem has been generalized from polynomial rings to other associative structures, such as Ore extensions. I will describe several further generalizations, in the form of non-associative versions of Hilberts basis theorem, for non-associative Ore extensions and related structures. An interesting asymmetry between the left and right versions of Hilberts basis theorem will appear, not present in the associative case. The talk is based on joint work with Per Bäck. (pdf)

In this talk, we give an overview of the theory of connections in noncommutative geometry, and discuss the existence of torsion free and metric compatible connections in the framework of derivation based noncommutative geometry. This is joint work with Victor Hildebrandsson. (pdf)

We exlore a necessary and sufficient condition for the existence of Levi-Civita connections in a derivation based calculus. We then derive a necessary Kähler condition on a fundamental form for finitely generated projective modules. We end with a brief discussion when this Kähler condition is also a sufficient condition. (pdf)

Determining whether a given noncommutative algebra admits a meaningful differential structure is both fundamental and challenging. In this talk, we contribute to this question by extending the classical differential geometric concept of a soldering form to the noncommutative setting. This also connects to ongoing research with Joakim Arnlind on Levi-Civita connections in the context of noncommutative principal bundles.

Generalized Weyl Algebras (GWAs) provide a flexible framework for studying various noncommutative algebras including the Weyl algebra, quantum groups, and Lie algebras. In this talk I will introduce GWAs and their representations, focusing on a new class of modules obtained by lifting free modules from the base ring. I will also describe the submodule structure of these modules, providing insights into composition series across numerous algebraic settings. Based on joint work with Samuel Lopes. (pdf)

This presentation is based on a master thesis finished in January 2024 at KTH. Starting with the work of Volker Strassen, algorithms for matrix multiplication have been developed which are time complexity-wise more efficient than the standard algorithm from the definition of multiplication. The general method of the developments has been viewing the bilinear mapping that matrix multiplication is as a three-dimensional tensor, where there is an exact correspondence between time complexity of the multiplication algorithm and tensor rank. The latter can be seen as a generalization of matrix rank, being the minimum number of terms a tensor can be decomposed as. However, in contrast to matrix rank there is no general method of computing tensor ranks, with many values being unknown for important three-dimensional tensors. To further improve the theoretical bounds of the time complexity of matrix multiplication, support rank of tensors has been introduced, which is the lowest rank of tensors with the same support in some basis. The goal of the master thesis was to go through the history of faster matrix multiplication, as well as specifically examining the properties of support rank for general tensors. In regards to the latter, a complete classification of rank structures of support classes was made for the smallest non-degenerate tensor product space in three dimensions. From this, the size of a support can be seen affecting the pool of possible ranks within a support class. At the same time, there is in general no symmetry with regards to support size occurring in the rank structures of the support classes, despite there existing a symmetry and bijection between mirrored supports. Discussions about how to classify support rank structures for larger tensor product spaces are also included. (pdf)

The simplification of algebraic expressions is a key step in many mathematical arguments; the systematic study of this is called rewriting, and the theory of Gröbner bases arises as an application to commutative algebra. For higher algebraic structures, things are not so straightforward; it may be quite difficult to distinguish one expression as simpler than another. Normally one requires the order which decides what is simpler (in Gröbner basis theory the “term order”) to be compatible with placing monomials in contexts, but if such contexts may permute free variables then this prohibits the order from distinguishing expressions which are equal up to such permutation. In for example the theory of Lie algebras, that would render all terms of both the anti-commutativity and the Jacobi axioms incomparable, and thus useless for rewriting. Unfailing Completion, introduced for term rewriting by Hsiang–Rusinowitch and Bachmair–Dershowitz–Plaisted in the 1980s, gets around this by abandoning compatibility, allowing the monomials to be ordered arbitrarily. Rewrite rules go every way, but are conditional: one may only apply a rule when doing so makes the expression smaller. This means one may need several resolutions of an ambiguity (critical pair), because what works in one context need not work in another. Enumeration of a sufficient set of cases is an interesting problem, with some similarities to that of enumerating all Gröbner bases of an ideal. In this talk I will show how to integrate unfailing completion for networks into my multi-sorted generic framework for diamond lemmas. (pdf)

In this talk, we will consider path rings, Cohn path rings, and Leavitt path rings associated with directed graphs, with coefficients in an arbitrary unital and associative ring R. For each of these types of rings, we will stipulate conditions on the graph that are necessary and sufficient to ensure that the ring satisfies the rank condition or the strong rank condition whenever R enjoys the same property. This talk is based on joint work with Karl Lorensen. (pdf)

The Graded Classification Conjecture states that for finite directed graphs $E$ and $F$, the associated Leavitt path algebras $L_\K(E)$ and $L_\K(F)$ are graded Morita equivalent, i.e., $\Gr L_\K(E) \approx_{\gr} \Gr L_\K(F)$, if and only if, their graded Grothendieck groups are isomorphic $K_0^{\gr}(L_\K(E)) \cong K_0^{\gr}(L_\K(F))$ as order-preserving $\mathbb Z[x,x^{-1}]$-modules. Furthermore, if under this isomorphism, the class $[L_\K(E)]$ is sent to $[L_\K(F)]$ then the algebras are graded isomorphic, i.e., $L_\K(E) \cong _{\gr} L_\K(F)$. In this talk we show that, for finite graphs $E$ and $F$ with no sinks and sources, an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism $K_0^{\gr}(L_\K(E)) \cong K_0^{\gr}(L_\K(F))$ gives that the categories of locally finite dimensional graded modules of $L_\K(E)$ and $L_\K(F)$ are equivalent, i.e., $\fGr[\mathbb{Z}] L_\K(E)\approx_{\gr} \fGr[\mathbb{Z}]L_\K(F).$ We further obtain that the category of finite dimensional (graded) modules are equivalent, i.e., $\fModd L_\K(E) \approx \fModd L_\K(F)$ and $\fGr L_\K(E) \approx_{\gr} \fGr L_\K(F)$. (pdf)

Given a set $A$ and an abelian group $B$ with operators in $A$, in the sense of Krull and Noether, we introduce the Ore group extension $B[x ; \sigma_B , \delta_B]$ as the additive group $B[x]$, with $A[x]$ as a set of operators. Here, the action of $A[x]$ on $B[x]$ is defined by mimicking the multiplication used in the classical case where $A$ and $B$ are the same ring. We derive generalizations of Vandermonde's and Leibniz's identities for this construction, and they are then used to establish associativity criteria. Additionally, we prove a version of Hilbert's basis theorem for this structure, under the assumption that the action of $A$ on $B$ is what we call weakly $s$-unital. Finally, we apply these results to the case where $B$ is a left module over a ring $A$, and specifically to the case where $A$ and $B$ coincide with a non-associative ring which is left distributive but not necessarily right distributive. This talk is based on joint work with Per Bäck, Johan Öinert and Johan Richter. (pdf)

In this talk I will introduce a generalization of n-hom-Lie algebras based on previous work with Jacobian hom-algebras. The Jacobi-type identity is asymmetrical, which leads to reexamine the axiom the adjoint is a hom-derivation that characterizes hom-Lie algebras, and to some extent their n-ary counterpart. I will introduce bi-generalized derivations as an approach to this problem and establish a relationship between them and the adjoint multiplication maps and introduce two types of extensions, one based on bi-generalized derivations and one based on generalized trace operators. (pdf)

This talk will be devoted to some constructions, examples, and open problems on color Hom-algebras.

Hsiang algebras are a class of nonassociative algebras defined in terms of a relation quartic in elements of the algebra. This class arises naturally in relation to the construction of real algebraic minimal cones. One can define a certain cubic form on Hsiang algebra A, analogues to the generic norm (or determinant) for Jordan algebras (the zero locus of the generic norm is a minimal cone in the ambient vector space). In my talk I will discuss some remarkable properties of the generic norm. In particular, for most Hsiang algebras A, the generic norm can be written in a certain orthonormal basis as the so-called Steiner triple form. Furthermore, the replication number of the corresponding triple system is uniquely determined by the Peirce dimensions of A. Some other applications will be discussed. The talk is based on an ongoing project with Prof. Daniel J. Fox (Madrid). (pdf)