# lie bracketの例文

- The Lie brackets are given by exactly the same formula as before.
- Coordinate basis vectors have the special property that their Lie brackets pairwise vanish.
- The Lie bracket is given by the commutator.
- This can be used to define the Lie bracket of vector fields as follows.
- Hence the Jacobi Lie bracket corresponds to the usual commutator for a matrix group:
- Where the omitted terms are known and involve Lie brackets of four or more elements.
- The space of vector fields forms a Lie algebra with respect to this Lie bracket.
- There are several approaches to defining the Lie bracket, all of which are equivalent.
- Thus, these two are consistency conditions for the Lie bracket on the tensor algebra.
- This observation generalises to the Lie bracket.
- The Lie bracket is defined by the formula
- This failure of closure under Lie bracket is measured by the " curvature ".
- Equivalently curvature can be calculated directly infinitesimally in terms of Lie brackets of lifted vector fields.
- The geometrical interpretation of the Lie bracket can be applied to the last of these equations.
- From the definition of the Lie bracket and with the use of Eq . 3 we have
- Antisymmetry follows from antisymmetry of the Lie bracket on and antisymmetry of the 2-cocycle.
- This is possible precisely because the tensor product is bilinear, and the Lie bracket is bilinear!
- Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields.
- By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.
- They are both ideals in \ mathfrak { d }, so the Lie bracket between them must vanjsh.