6.8. Lie brackets

The Lie bracket of two vector fields is a third vector field.

Using abstract indices a bracket can be expressed as

In[509]:=

DefTensor[x[a], M3]

** DefTensor: Defining tensor x[a] .

In[510]:=

Bracket[a][w[b], x[b]]

Out[510]=

[w_ ^b, x_ ^b]^a

Note that the indices of the vector fields are ultraindices. The index of the resulting vector field is represented outside the bracket.

It has almost all expected properties: bilinearity, antisymmetry, derivation, ..., but not yet the Jacobi rule.

In[511]:=

Bracket[a][x[b] + w[b], 3w[b] + r[] x[b]]

Out[511]=

-3 [w_ ^b, x_ ^b]^a + r_^ [w_ ^b, x_ ^b]^a + x_ ^a ∂_w^ r_^ + x_ ^a ∂_x^ r_^

In[512]:=

UndefTensor/@{x, w} ;

** UndefTensor: Undefined tensor x

** UndefTensor: Undefined tensor w

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