In 1982, Robert Griess provided the first construction of the Monster simple group M as a group of automorphisms of a 196884-dimensional commutative nonassociative algebra V. In subsequent years, this construction was simplified and analyzed in a number of ways. In particular, John Conway discovered an association between a distinguished set of idempotents (called axes) in V and a conjugacy class of involutions in M. In 1996, Simon Norton studied the subalgebras in V generated by a few axes. Alexander Ivanov axiomatized the Griess-Conway-Norton algebra as a particular instance of the so-called Majorana algebras. Hall, Shpectorov and Rehren extended the above results for general axial algebras. On the other hand, it has been recently realized that there is a class of axial algebra which naturally appears in a completely different area: minimal (i.e. zero mean curvature) cones. In my talk I try to explain some remarkable connections between group theoretic, algebraic and geometrical parts of the above picture. (pdf)

We introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on almost Kähler manifolds. We first give a short review of Lie-Rinehart algebras, and after that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts; e.g. curvature and Levi-Civita connections. Several examples are provided in order to illustrate the novel concepts. (pdf)

Skew inverse semigroup rings were introduced by Exel and Vieira in 2010 and are generalizations of partial skew group rings. In a joint work with Beuter, Goncalves and Royer, I have obtained necessary and sufficient conditions for the simplicity of certain skew inverse semigroup rings. During my talk I will review the construction of a skew inverse semigroup ring as well as our results. As an application, I will briefly mention how our results can be used to give a new proof of the simplicity criterion for Steinberg algebras. Steinberg algebras are algebraic versions of Renault's C*-algebras of groupoids and include e.g. all Leavitt path algebras. (pdf)

We obtain sufficient criteria for simplicity of systems, that is, rings $R$ that are equipped with a family of additive subgroups $R_s$, for $s \in S$, where $S$ is a semigroup, satisfying $R = \sum_{s \in S} R_s$ and $R_s R_t \subseteq R_{st}$, for $s,t \in S$. These criteria are specialized to obtain sufficient criteria for simplicity of, what we call, s-unital epsilon-strong systems, that is systems where $S$ is an inverse semigroup, $R$ is coherent, in the sense that for all $s,t \in S$ with $s \leq t$, the inclusion $R_s \subseteq R_t$ holds, and for each $s \in S$, the $R_s R_{s^*}$-$R_{s^*} R_s$-bimodule $R_s$ is s-unital. As an aplication of this, we obtain generalizations of recent sufficient criteria for simplicity of skew inverse semigroup rings, by Beuter, Goncalves, Öinert and Royer, and then, in turn, for Steinberg algebras by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michel. (pdf)

The class of epsilon-strongly graded rings was recently introduced by P. Nystedt, J. Öinert and H. Pinedo as a generalization of strongly graded rings. In this talk, I will present some background and examples and then discuss characterizations of noetherian and artinian epsilon-strongly graded rings. We will apply the general results to the important special cases of unital partial crossed products and Leavitt path algebras of a finite graph. (pdf)

The notion of a free action of a quantum group on a C∗-algebra provides a natural framework for noncommutative principal bundles, which are not of purely mathematical interest only. Indeed, besides their structure theory and relation with K-theory, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry and mathematical physics. In this talk we explain how to systematically study geometric aspects of noncommutative principal bundles by means of Connes’ spectral triples, which allow to extend many techniques from Riemannian spin geometry to the noncommutative setting. (pdf)

In a series of papers we have developed a framework, named Pseudo-Riemannian calculi, to discuss Levi-Civita connections on vector bundles (projective modules) over noncommutative algebras. It can be shown that at most one torsion-free and metric connection exists, and under certain reality conditions, the curvature operator has the same symmetries as in the classical situation. In this talk, I will give an overview of these results and in particular discuss their application to the noncommutative torus and the noncommutative 3-sphere and a noncommutative version of the Chern-Gauss-Bonnet theorem. (pdf)

The q-derivations on a noncommutative three sphere are given. In this paper we show that it is possible to construct a Levi-Civita connection based on these q-derivations. Such a connection is constructed on a finitely projective module in general with a hermitian form. By specifying to the module of differential forms over the noncommutative sphere, we can find a metric and torsion free connection where the torsion freeness is defined appropriately. (pdf)

n this talk an introductory overview on the subject of Hom-algebra structures will be given with emphasize on hom-algebra generalizations of Lie algebras and associative algebras. In 1990’th in the pioneering works of Curtright, Zachos, Ellinas, Chaichian, Kulish, Lukierski, Presnajder, Popowicz, Isaev, Aizawa, Sato, Hu quantum deformations of algebras, q-deformed oscillator algebras, q-deformations of Witt and Virasoro algebras and related families of algebras defined by generators and parameter commutation relations have been constructed in connection to quantum deformations and discretized models of mechanics and quantum mechanics, q-deformations of vertex operators, q-deformed conformal quantum field theory, q-deformed integrable systems, q-deformed superstrings and central extensions. Also various quantum n-ary extensions of Numbu mechanics and related n-ary extensions of differential structures and of Lie algebras Jacobi identities have been considered. It was noticed in particular that many of quantum algebras and q-deformed Lie algebras obey certain q-deformed versions of Jacobi identity generalizing Lie algebras Jacobi identity. Motivated by these works Hartwig, Larsson and Silvestrov in 2003 developed a general method of obtaining such deformations and generalized Jacobi identities based on general twisted derivations. This development, as well generalizations of supersymmetry, lead to development of more general algebraic structures such as quasi-Lie algebras and Hom-Lie algebras, Hom-associative and Hom-Lie admissible algebras, Hom-Jordan algebras, Hom-Poisson algebras, Hom-Yang-Baxter equations, Hom-bialgebras, Hom-Hopf algebras, and other hom-algebra structures, as well as Hom-Nambu and Hom-Nambu Lie algebras, n-ary Hom-algebra generalizations of Nambu algebras, associative algebras and Lie algebras and methods for their constructions and classifications. (pdf)

Rather than presenting new research (at least not primarily), this talk is an introduction to some techniques from the borderlands between Algebra, Discrete Mathematics, and Computer Science — demonstrating how they can be used to gain insights into questions of more traditional algebra. Topics that may be mentioned are the power series formalism for formal languages, regular languages (word and tree variants), context-free languages, determinism, and Hilbert series. Examples may include associative algebras (near-commutative vs. more general), non-associative algebras (particularly hom-algebras), and operads. (pdf)

Let F be a field, and fix an element q of F. The q-deformed Heisenberg algebra H(q) is the unital associative algebra over F with generators A, B and relation which asserts that AB - qBA is the unity element. By the set of all Lie polynomials or undeformed commutators in A, B, we mean the Lie subalgebra L of H(q) generated by A, B. We present results from several studies that describe such Lie polynomials. We fully describe L in terms of a basis and a corresponding commutator table, which vary according to cases based on the parameter q. If F is the complex field, and if q is in the interval (0,1), then H(q) is isomorphic to an algebra of operators on some sequence space such that all the compact operators are in L, and the image of H(q) in the Calkin algebra is an algebra of Laurent polynomials in one indeterminate. (pdf)

I will review the definition of Ore extensions, a non-commutative (associative) generalization of polynomial rings. I will present some results on when Ore extensions are simple. I will then discuss non-associative Ore extensions, which is a new concept, and present some theorems on when they are simple, generalizing the results for associative Ore extensions. The part about non-associative Ore extensions is based on joint work by Patrik Nystedt, Johan Öinert and myself. (pdf)

I will give a brief introduction to non-associative, non-commutative polynomial rings known as hom-associative Ore extensions. Within this framework, we will then see, for example, how the first Weyl algebras, the quantum planes, and Hilbert's basis theorem can be generalized into non-associative versions. (pdf)

For any n-dimensional hom-Lie algebra, a system of polynomial equations is obtained from the hom-Jacobi identity, having structure constants of both the skew-symmetric bilinear map and twisting linear endomorphism. These equations are expressed as linear system in structure constants of the twisting linear endomorphism and thus represented as linear maps. We give a description of realization of three-dimensional and four-dimensional hom-Lie algebras, for kernel solutions of the minimum dimension. We further give some examples of four- dimensional Hom-Lie algebras constructed from a general nilpotent linear endomorphism. (pdf)

The main object of this talk is the multi-parametric family of algebras generated by $Q$ and the set $\{S_j\}$ satisfying commutation relations of the form $S_jQ=\sigma_j(Q)S_j$ where $\sigma_j$ is some function for which the expression $\sigma_j(Q)$ makes sense. Through a linear transformation of the $S_j$ generators, we introduce and consider another equivalent multi-parametric family of algebras. We derive general reordering formulas for these algebras and apply the results to the description of commutative subalgebras. We give operator representations for particular cases of $\sigma_j$, and in particular, we show that some multi-parameter deformed symmetric difference and multiplication operators satisfy defining relations of the corresponding algebras. Finally we consider the algebras also in the context of $(\sigma,\tau)$-derivations and Ore extensions. (pdf)