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In http://en.wikipedia.org/wiki/Lie_algebra, Lie bracket operation is defined as having bilinearity: $[ax+by,z] = a[x,z]+b[y,z], [z,ax+by] = a[z,x]+b[z,y]$.

But in http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields, it is said as the following: "for functions f and g, $[fX,gY] = fg[X,Y] + fX(g)Y-gY(f)X$." ($X,Y$ vector fields.)

My question is, the later description seems to violate bilinearity - function would anyway be evaluated at $p$ of smooth manifold $M$ which would make situation basically similar to the first description. Am I wrong here?

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From what is written there, I can't figure out what is meant by $fX$. It is not even explained what $f$ is other than "a function". Could $f$ be from $M$ to $\Bbb R$? Then $fX(m):=f(m)X(m)$ would make sense... No doubt someone will recognize what is meant. –  rschwieb Apr 10 '13 at 13:25

1 Answer 1

Vector fields form a Lie algebra over the real numbers, not the ring of functions (which is not a field anyways). If $a,b$ are real numbers and $X$ and $Y$ vector fields then $[aX,bY] = ab[X,Y]$.

Note that this is consistent with what you wrote for functions if you consider $a$ and $b$ to be constant functions, since then $X(b) = 0 = Y(a)$.

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