Re: What is a cocycle in the context of lie algebras?

On 15-02-2007 0:46, smetalman2006@xxxxxxxxx wrote:

Please explain in simple terms that a physicist would be able to
understand. Thanks,

Let _g_ be a Lie algebra which acts on a vector space V. In other
words, suppose that for each X in _g_ and each vector _v_ in V, there
is a vector X.v in V such that

X.(Y.v) - Y.(X.v) = [X,Y].v.

Then a cocycle is a map _w_ from _g_ x _g_ into V such that:

1) w(X_1 + X_2,Y) = w(X_1,Y) + w(X_2,Y);

2) if _x_ is a number, then w(x*X,Y) = x*w(X,Y);

3) w(X,Y) = -w(Y,X);

4) w([X,Y],Z) - w([X,Z],Y) + w([Y,Z],X) =

= X.w(Y,Z) - Y.w(X,Z) + Z.w(X,Y).

A way of building cocycles is as follows: let _f_ be a linear
map from _g_ into V and define

w(X,Y) = f([X,Y]) - X.f(Y) + Y.f(X).

Best regards,

Jose Carlos Santos

.

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