Following the spirit of a recent work of one of the authors (J. Phys. A: Math. Theor. 44 (2011) 045301), the essential structure of the generalized Pauli group of a qubit-qudit, where d = 2 and an integer k ≥ 2, is recast in the language of a finite geometry. A point of such geometry is represented by the maximum set of mutually commuting elements of the group and two distinct points are regard...

This paper proves the Commuting Derivations Conjecture in dimension three: if D1 and D2 are two locally nilpotent derivations which are linearly independent and satisfy [D1, D2] = 0 then the intersection of the kernels, A1 ∩ A2 equals C[f ] where f is a coordinate. As a consequence, it is shown that p(X)Y + Q(X, Z, T ) is a coordinate if and only if Q(a, Z, T ) is a coordinate for every zero a ...

Let G be a non-abelian group. The non-commuting graph AG of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with ω(AG) ≤ 57...

Given a non-abelian finite group G, let π(G) denote the set of prime divisors of the order of G and denote by Z(G) the center of G. The prime graph of G is the graph with vertex set π(G) where two distinct primes p and q are joined by an edge if and only if G contains an element of order pq and the non-commuting graph of G is the graph with the vertex set G−Z(G) where two non-central elements x...

We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. We define an abstract Hermitian algebra (AH-algebra) to be the directed group of an e-ring that contains a semitransparent element, has the quadratic annihilation property, and satisfies a Vigier condition on pairwise commuting ascending sequences. All of this terminology is expl...

let $mathcal {a} $ and $mathcal {b} $ be c$^*$-algebras. assume that $mathcal {a}$ is of real rank zero and unital with unit $i$ and $k>0$ is a real number. it is shown that if $phi:mathcal{a} tomathcal{b}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $phi(|a|^k)=|phi(a)|^k $ for all normal elements $ainmathcal a$, $phi(i)$ is a projection, and there exists a posit...

Let G be an arbitrary discrete group and let T = C[G] be its group algebra over the complex numbers C. If 9? is an irreducible representation of the algebra then 9i(r)=P is primitive and hence isomorphic to a dense set of linear transformations over D, the commuting ring of 9J [4, p. 28]. Let L be the center of D. If dimr, P < °o then we say that 9? is finite and since P is central simple over ...

The commuting graph of a group G, denoted by Γ(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Zm be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Zm) to contribute to the conje...

Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, its quotient under adjoint action, $Rep(\mathbb{Z}^n,G):=Hom(\mathbb{Z}^n,G)/G$. In this article we study many cases compute homotopy groups $\pi_2(Hom(\mathbb{Z}^n,G))$. For simply--connected simple show that $\pi_2(Hom(\mathbb{Z}^2,G))\cong ...