in this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid g(n) . also we show that g(n) contains commuting regular subsemigroup and give a necessary and sucient condition for the groupoid g(n) to be commuting regular.

A master equation (n dimensional non–Abelian current conservation law with mutually commuting current components) is introduced for multi-dimensional non-linear field theories. It is shown that the master equation provides a systematic way to understand 2-d integrable non-linear equations as well as 4-d self-dual equations and, more importantly, their generalizations to higher dimensions.

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N -qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized q...

We describe new Banach (not C∗ !) algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit ball B, where n > 2. For n = 2 all these algebras collapse to the single C∗-algebra generated by Toeplitz operators with quasi-parabolic symbols. As a by-product, we describe the situations when the product of mutually commuting Toeplitz operators is a To...

The natural su(N) generalization of the XX model is introduced and analyzed. It is defined in terms of the characterizing properties of the usual XX model: the existence of two infinite sequences of mutually commuting conservation laws and the existence of two infinite sequences of mastersymmetries. The integrability of these models, which cannot be obtained in a degenerate limit of the su(N)-X...

Using the spectral subspaces obtained in [HS], Brown’s results (cf. [Bro]) on the Brown measure of an operator in a type II1 factor (M, τ) are generalized to finite sets of commuting operators in M. It is shown that whenever T1, . . . , Tn ∈ M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure μT1,...,Tn on B(C) such that for all α1, . ...

We investigate local conserved charges in principal chiral models in 1+1 dimensions. There is a classically conserved local charge for each invariant tensor of the underlying group. We prove that these are always in involution with the non-local Yangian charges and we study their classical Poisson bracket algebra. For the groups U(N), O(N) and Sp(N) we show that there are infinitely many commut...

A three-dimensional polynomial algebra of order m is defined by the commutation relations [P0, P±] = ±P±, [P+, P−] = φ (P0) where φ(P0) is an m-th order polynomial in P0 with the coefficients being constants or central elements of the algebra. It is shown that two given mutually commuting polynomial algebras of orders l and m can be combined to give two distinct (l + m + 1)-th order polynomial ...

Proof. For two commuting matrices A and B, each eigenspace of A is invariant under B. Indeed, if (A−λIn)v = 0 then (A−λIn)Bv = B(A−λIn)v = B(0) = 0. For n = 1, each element of F is a complex number; thus, the statement of the lemma is true. Suppose the statement of the lemma is true for all n < m for some m ≥ 2. Let F be a set of mutually commuting diagonalizable matrices in Mm(C). Take A ∈ F ....