Lie Algebra: Why is 2D Nilpotent Lie Algebra Abelian?

[lion8172]

I read in Knapp's book on Lie algebras that "a 2-dimensional nilpotent Lie algebra is abelian." Why is this the case? Can somebody who knows please tell me?

 

[morphism]

How many 2-dimensional Lie algebras are there (up to isomorphism)? There are two: one's abelian and the other's not. Try proving that the nonabliean one cannot be nilpotent.

 

[tacman]

A Lie algebra is considered nilpotent if there exists a finite positive integer k such that the kth iteration of the Lie bracket operation on any two elements of the algebra results in the zero element. In other words, the Lie bracket operation "kills" all elements after k iterations.

In a 2-dimensional nilpotent Lie algebra, there are only two basis elements, say x and y. Let's consider the Lie bracket [x,y] which is defined as the commutator of x and y, i.e. [x,y]=xy-yx. Since the Lie algebra is nilpotent, we know that [x,y] must equal zero after a certain number of iterations, say k. This means that [x,y] is a linear combination of x and y, and hence can be written as a linear combination of the basis elements. Therefore, we can write [x,y] as aX + bY, where a and b are constants and X and Y are the basis elements.

Now, since [x,y] is a linear combination of x and y, we can use the Jacobi identity (which is satisfied by all Lie algebras) to write [x,[x,y]] and [[x,y],y] in terms of x and y. By the nilpotency condition, we know that these two terms must also equal zero after a certain number of iterations. This leads to a system of equations:

aX + bY = 0
-aX + 2bY = 0

Solving this system of equations, we get a=b=0. This means that [x,y] must equal zero, and hence the Lie algebra is abelian.

In summary, the fact that a 2-dimensional nilpotent Lie algebra is abelian is a consequence of the nilpotency condition and the Jacobi identity. It can also be seen as a result of the simplicity of the algebra, where there are only two basis elements and the Lie bracket operation can only result in a linear combination of these elements.