Search results for "Commutator"

Normalities and Commutators

2010

We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and we define a new notion of normality in terms of this commutator. Our main result is to extend to any semi-abelian category the following well-known characterization of normal subgroups: a subobject K is normal in A if. and only if, {[A, K] <= K. (C) 2010 Elsevier Inc. All rights reserved.}

Commutator Equations and the Equationally-Defined Commutator

2015

If \(\underline{s} = s_{1},\ldots,s_{m}\), and \(\underline{t} = t_{1},\ldots,t_{m}\) are sequences of terms (both sequences of the same length m) and X is a set of equations then $$\displaystyle{\underline{s} \approx \underline{ t} \in X}$$ abbreviates the fact that s i ≈ t i ∈ X for i = 1, …, m.)

Linear Methods in Nilpotent Groups

1982

The subject of this chapter is commutator calculation. It will be recalled that the commutator [x, y] of two elements x, y of a group is defined by the relation $$ [x,y] = {{x}^{{ - 1}}}{{y}^{{ - 1}}}xy. $$ . We then have $$ [xy,z] = {{[x,z]}^{y}}[y,z],\quad [x,yz] = [x,z]{{[x,y]}^{z}}. $$ . These relations are rather similar to the conditions for bilinearity of forms, and there are a number of ways of formalizing this similarity. Once this is done, commutator calculations can be done by linear methods. Several examples of theorems proved by this method will be given in this chapter.

Do metric independent classical actions lead to topological field theories?

1991

Abstract We investigate the quantum theory of non-abelian BF -systems (gauge theories with the classical metric independent action ∫ tr BF A ). The fact that due to a complicated (on-shell reducible) gauge structure the quantum action of these theories does not differ only by a BRST commutator from the classical action, and that moreover the BRST operator turns out to be metric dependent, renders the standard arguments for metric independence inapplicable. We establish the topological nature of these models and argue that in gauge theories the information on gauge invariance is contained entirely in the metric independent part of the BRST operator. We make some general remarks on the relati…

Neutrinoless double beta decays of 106Cd revisited

2011

Abstract Neutrinoless double beta ( 0 ν 2 β ) decays of 106 Cd are studied for the transitions to the ground state, 0 gs + , and 0 + excited states in 106 Pd by using realistic many-body wave functions calculated in the framework of the quasiparticle random-phase approximation and its extensions. Effective, G-matrix-based nuclear forces are used in large single-particle model spaces. Both the β + β + and β + EC channels of the 0 ν 2 β decay are discussed and half-lives are computed. Particular attention is devoted to the study of the detectability of the resonant neutrinoless double electron capture ( R 0 ν ECEC ) process in 106 Cd. The calculations of the present article constitute the thu…

On the Leibniz bracket, the Schouten bracket and the Laplacian

2003

International audience; The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them is obtained. Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie algebra structure. In particular, those algebras generated by the Leibniz bracket of the divergence and the Laplacian operators on the exterior algebra are considered, and the expression of the Laplacian for the product of two functions is generalized for arbitrary exterior forms.

Tripartite separability conditions exponentially violated by Gaussian states

2014

Starting with a set of conditions for bipartite separability of arbitrary quantum states in any dimension and expressed in terms of arbitrary operators whose commutator is a $c$-number, we derive a hierarchy of conditions for tripartite separability of continuous-variable three-mode quantum states. These conditions have the form of inequalities for higher-order moments of linear combinations of the mode operators. They enable one to distinguish between all possible kinds of tripartite separability, while the strongest violation of these inequalities is a sufficient condition for genuine tripartite entanglement. We construct Gaussian states for which the violation of our conditions grows exp…

Weak commutation relations of unbounded operators: Nonlinear extensions

2013

We continue our analysis of the consequences of the commutation relation $[S,T]=\Id$, where $S$ and $T$ are two closable unbounded operators. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. {We also consider what we call, adopting a physical terminology}, a {\em nonlinear} extension of the above commutation relations.

Additivity of the Equationally-Defined Commutator and Relatively Congruence-Distributive Subquasivarieties

2015

Additivity of the Equationally-Defined Commutator

2015

In this chapter we are concerned with the problem of additivity of the equationally defined commutator.