‎‎Let $R$ be a non-commutative ring with unity‎. ‎The commuting graph‎ of $R$ denoted by $Gamma(R)$‎, ‎is a graph with a vertex set‎ ‎$Rsetminus Z(R)$ and two vertices $a$ and $b$ are adjacent if and only if‎ $ab=ba$‎. ‎In this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. It is shown that‎, ‎$Gamma(R)$ is the disjoint ...

The Maxwell theory on non-commutative spaces has been considered. The non-linear equations of electromagnetic fields on non-commutative spaces were obtained in the compact spin-tensor (quater-nion) form. We found the symmetric energy-momentum tensor and its non-zero trace. So, the trace anomaly of the energy-momentum tensor was obtained in electrodynamics on non-commutative spaces. It was noted...

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our ”strict” Positivstellensatz is positive news, on the opposite extreme fro...

We present several results concerning non-commutative instantons and the Seiberg–Witten map. Using a simple ansatz we find a large new class of instanton solutions in arbitrary even dimensional non-commutative Yang–Mills theory. These include the two dimensional “shift operator” solutions and the four dimensional Nekrasov–Schwarz instantons as special cases. We also study how the Seiberg–Witten...

We examine the unitarity properties of spontaneously broken non-commutative gauge theories. We find that the symmetry breaking mechanism in the non-commutative Standard Model of Chaichian et al. leads to an unavoidable violation of tree-level unitarity in gauge boson scattering at high energies. We then study a variety of simplified spontaneously broken non-commutative theories and isolate the ...

We characterise algebras commutative with respect to a Yang-Baxter operator (quasi-commutative algebras) in terms of certain cosimplicial complexes. In some cases this characterisation allows the classification of all possible quasi-commutative structures.

We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for ...

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...

Classical differential geometry can be encoded in spectral data, such as Connes’ spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes’ non-commutative spin geometry encompassing noncommutative Riemannian, symplectic, complex-Hermitian and (Hyper-) Kähler geometry...