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aalma

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Misplaced Homework Thread

-Verify that the space ##Vect(M)## of vector fields on a manifold ##M## is a Lie algebra with respect to the bracket.

-More generally, verify that the set of derivations of any algebra ##A## is a Lie algebra with respect to the bracket defined as ##[δ_1,δ_2] = δ _1◦δ_2− δ_2◦δ_1##.

In the first part of the question am I supposed to show that the bracket operation satisfies the axioms of a Lie algebra: bilinearity, skew-symmetry, and the Jacobi identity? Given ##X, Y \in Vect(M)## we look at ##[X,Y]f=XYf-YXf## for ##f \in C^{\infty}(M)##, and then check the axioms?

But is not it trivial knowing that ##Vect(M)## has a structure of a vector space on ##R## and if ##f \in C^{\infty}(M), X \in Vect(M)## then ##fX \in Vect(M)## ?

I would be glad if you can tell me how do these parts work.

Thank you for help

-More generally, verify that the set of derivations of any algebra ##A## is a Lie algebra with respect to the bracket defined as ##[δ_1,δ_2] = δ _1◦δ_2− δ_2◦δ_1##.

In the first part of the question am I supposed to show that the bracket operation satisfies the axioms of a Lie algebra: bilinearity, skew-symmetry, and the Jacobi identity? Given ##X, Y \in Vect(M)## we look at ##[X,Y]f=XYf-YXf## for ##f \in C^{\infty}(M)##, and then check the axioms?

But is not it trivial knowing that ##Vect(M)## has a structure of a vector space on ##R## and if ##f \in C^{\infty}(M), X \in Vect(M)## then ##fX \in Vect(M)## ?

I would be glad if you can tell me how do these parts work.

Thank you for help