# Lie bracket

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## Lie bracket

[′lē ‚brak·ət]
(mathematics)
Given vector fields X, Y on a manifold M, their Lie bracket is the vector field whose value is the difference between the values of XY and YX.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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The author constructs optimal paths as one parameter groups defined in terms of matrix exponentiation, proves matrix groups are manifolds, explores the properties of the Lie bracket, and uses maximal tori to prove several theorems about familiar compact matrix groups.
A unicycle model or a two-wheel car model, shown in Figure 1, is basically a three-dimensional nonholonomic system having two inputs and three states with depth-one Lie bracket. A two-wheel car kinematic model is defined as 
The usual Lie bracket on vector fields induces the bracket on [GAMMA](A), and the anchor is given by [rho] = dt: A [right arrow] TM.
where the Lie bracket on Gr(G) is given by the group commutator.
Recall that a Lie bracket on a vector space V is a bilinear binary product [*, *]: V x V [right arrow] V such that for all x, y, z [member of] V,
The tangent space of G at identity is the Lie algebra g, where the Lie bracket is defined.
To compute the Lie Bracket amounts to compute the Lie Bracket over [[universal].sub.x[member of]u] [L.sub.x]M U [subset or equal to] M.
Given two polynomials P, Q [member of] K<A>, their Lie product is the Lie bracket [P, Q] = PQ - QP.
Our convention is that the Lie bracket IX, Y] of vector fields on a manifold M is given by [X, Y]f = Y(Xf) -X(Yf) for f [element of] [C.sup.[infinity]](M).
The notion of semidirect sums [bar.g] = g [??] [g.sub.c] means that g and [g.sub.c] satisfy [mathematical expression not reproducible] denoting the Lie bracket of [bar.g].
The tangent bundle (TM,t,M) with the usual Lie bracket and [rho] equal to the identity map form a Lie algebroid.
An algebroid is an ALS ([theta],D, [[*,*].sub.[theta]]) which enjoys the property that [DX,DY ] = D[[X, Y ].sub.[theta]], ([for exists])X, Y [member of] [GAMMA]([theta]), where the first bracket is the Lie bracket on X(M).

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