# Bracket Operation

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### C.h. Moog – 1st expert on this subject based on the ideXlab platform

• ##### On linear equivalence for time-delay systems
Proceedings of the 2010 American Control Conference, 2010
Co-Authors: C. Califano, L.a. Marquez-martinez, C.h. Moog

Abstract:

The aim of the present paper is to introduce new mathematical tools for the analysis and control of nonlinear time-delay systems (NLTDS). An Extended Lie Bracket Operation equivalent to the Lie Bracket Operation for system without delays is introduced. It will be shown that this Operation, which generalizes that introduced in, helps to characterize certain properties of a given submodule, such as nilpotency. This basic property is then used to define the conditions under which a given unimodular matrix represents a bicausal change of coordinates. The effectiveness of the proposed approach will be shown by solving an important basic problem: to characterize if a NLTDS is equivalent or not, to a Linear Time-Delay System by bicausal change of coordinates.

### H. Barcelo – 2nd expert on this subject based on the ideXlab platform

• ##### On Some Submodules of the Action of the Symmetrical Group on the Free Lie Algebra
Journal of Algebra, 1993
Co-Authors: H. Barcelo, S. Sundaram

Abstract:

Abstract The free Lie algebra Lie[ A ] over the complex held, on an alphabet A , is the smallest subspace of the complex linear span of all words in A , which is closed under the Bracket Operation [ u , v ] = uv − vu . Define Lie n to be the subspace of the free Lie algebra Lie[1, …, n ] spanned by Bracketings consisting of words which are permutations of {1, …, n }. The symmetric group S n acts on Lie n by replacement of letters, giving an ( n − 1)!-dimensional representation isomorphic to the induction ω↑ S n C n , where C n is the cyclic group of order n and ω is a primitive n th root of unity. Bracketings in Lie n may be represented graphically by labelled binary trees with n leaves. Fix a particular unlabelled binary tree T ; then the vector subspace spanned by all words corresponding to the n ! possible labellings of T is an S n -module V T . In this paper we study the representations afforded by certain classes of trees T . We show that the plethysm V S [ V T ] is isomorphic to the submodule corresponding to a tree S [ T ] which has a natural description in terms of the trees S and T .

• ##### On the action of the symmetric group on the free lie algebra and the partition lattice
Journal of Combinatorial Theory Series A, 1990
Co-Authors: H. Barcelo

Abstract:

Abstract The Free Lie Algebra over an alphabet A , denoted here by LIE[ A ], is the smallest subspace of the linear span of the A -words which contains the letters and is closed under the Bracket Operation [ f , g ] = fg − gf . A permutation σ acts on words by replacing each occurrence of the letter a i by a σ i . This action linearly extends to LIE[ A ]. We are concerned here with the action of the symmetric group S n on the subspace of LIE[ A ] which is the linear span of Bracketings of words which are permutations of the letters of the alphabet. It follows from the work of Hanlon, Stanley, and Joyal that this action and the action of S n on the top homology of the partition lattice Π n induce similar representations (up to tensoring with the alternating character). It follows from the work of Garsia and Stanton that the action on the homology is similar to the action on a suitably defined top portion of the Stanley-Reisner ring. In this paper we derive a direct combinatorial proof of the similarity of these three actions by choosing natural bases in each of these three spaces and comparing the matrices corresponding to the simple reflections.

### Alan Weinstein – 3rd expert on this subject based on the ideXlab platform

• ##### Plasma in monopole background is not twisted Poisson.
arXiv: Mathematical Physics, 2019
Co-Authors: Manuel Lainz, C. Sardón, Alan Weinstein

Abstract:

For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding Bracket Operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson Bracket; however, it is twisted Poisson in the sense that the Jacobiator comes from a closed 3-form.
The space $\mathcal D$ of densities on phase space is the state space of a plasma. The twisted Poisson Bracket on phase-space functions gives rise to a Bracket on functions on $\mathcal D$. In the absence of monopoles, this is again a Poisson Bracket. It has recently been shown by Heninger and Morrison that this Bracket is not Poisson when monopoles are present. In this note, we give an example where it is not even twisted Poisson.

• ##### Omni-Lie Algebras
arXiv: Representation Theory, 1999
Co-Authors: Alan Weinstein

Abstract:

We show that the space R^n x gl(n,R) with a certain antisymmetric Bracket Operation contains all n-dimensional Lie algebras. The Bracket does not satisfy the Jacobi identity, but it does satisfy it for subalgebras which are isotropic under a certain symmetric bilinear form with values in R^n. We ask what the corresponding “group-like” object should be. The Bracket may be obtained by linearizing at a point the Bracket on TM + T*M introduced by T. Courant for the definition of Dirac structures, a notion which encompasses Poisson structures, closed 2-forms, and foliations.