What is the Dimension of Lie Algebras \mathfrak{g} and \mathfrak{h}?

[Ted123]

Homework Statement

$\mathfrak{g}$ is the Lie algebra with basis vectors $E,F,G$ such that the following relations for Lie brackets are satisfied:

$[E,F]=G,\;\;[E,G]=0,\;\;[F,G]=0.$

$\mathfrak{h}$ is the Lie algebra consisting of 3x3 matrices of the form

$\begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{bmatrix}$ where $a,b,c$ are any complex numbers. The vector addition and scalar multiplication on $\mathfrak{h}$ are the usual operations on matrices.

The Lie bracket on $\mathfrak{h}$ is defined as the matrix commutator: $[X,Y] = XY - YX$ for any $X,Y \in \mathfrak{h}.$

The Attempt at a Solution

Is the dimension of $\mathfrak{g}$ and $\mathfrak{h}$ both 3?

 

[lippyka]

Yes, the dimension of \mathfrak{g} and \mathfrak{h} are both 3. This is because they each have three basis vectors or matrices, respectively.