Is there any good example about Lie algebra homomorphisms? My textbook gave an example of the trace, but I think to get a better comprehension, more examples are still needed.
Any example will be helpful ~
 A: A good source of examples is the free Lie algebra $\mathcal L(n)$, on generators $X_1,\ldots, X_n$. This is defined as the vector space with basis given by all formal bracketing expressions of generators, such as $[X_1,X_2]$, $[X_3+2X_4,[X_5,X_7]]$, etc. One the takes the quotient by relators representing antisymmetry of the bracket, the Jaobi identity and multilinearity.
This gives a lot of homomorphisms. For example, $\mathcal L(n)$ maps homomorphically onto $\mathcal L(n-1)$ where the homomorphism is defined by setting a variable equal to $0$. Similarly $\mathcal L(n)$ maps to $\mathcal L(n+1)$ by inclusion, which is a Lie algebra homomorphism. Indeed, since the free Lie algebra only includes relations that are present in every Lie algebra, one can show that there is a surjective homomorphism from a free Lie algebra onto any finitely generated Lie algebra.
Another example that just occurred to me: any associative ring can be thought of as a Lie algebra using the bracket $[a,b]=ab-ba$. Any ring homomorphism will induce a Lie algebra homomorphism.
A: Let $V_k$ be the space of homogeneous polynomials of degree $k$ on the unknowns $x_1,x_2,\ldots,x_n$. The differential operators $a_{ij}=x_i\dfrac{\partial}{\partial x_j}$ act on the space $V_k$. It is an easy exercise to show that
$$
[a_{ij},a_{kl}]=\delta_{jk}a_{il}-\delta_{il}a_{kj}.
$$
Therefore the mapping $e_{ij}\mapsto a_{ij}$ is a homomorphism of Lie algebras from $gl_n$ to $gl(V_k)$.
Addendum: Here $\delta_{ij}$ is the Kronecker delta, and by $e_{ij}$ I mean the matrix with 1 at the intersection of row $i$ and column $j$ and zeros elsewhere. This is, of course, an example of a simple representation of $gl_n$, but I hope it will be helpful also, when you reach that point in your studies.
