# Isomorphic representations of a Lie Algebra

The left-invariant vector fields on a Lie Group $G$ constitute a canonical Lie Algebra $\mathfrak{g}$ for $G$, Lie($G$).

With some adequate definition, one can also define a bracket on a vector space such that one obtains a Lie Algebra, e.g. commuting matrices for $\mathfrak{gl}(n)$, and in this case Lie($GL(n)$)$\simeq \mathfrak{gl}(n)$.

How can one find (describe) all possible isomorphic Lie Algebras to the canonical one, Lie($G$)?

• Describe isomorphic Lie Algebras? I don't understand the question. It's like asking "find/describe all sets equinumerous with a given set $X$". Or are you saying "describe all Lie groups with a Lie algebra isomorphic to the given one"? – freakish Oct 10 '17 at 9:41
• Once you've stripped away any of the extra structure on $\mathrm{Lie}(G)$ (for example, the fact that the vector space is made up of vector fields) and only remembered the vector space and the Lie bracket, there's no way you would know if you had the canonical one or not. – Joppy Oct 10 '17 at 9:57
• @freakish For instance, $\mathfrak{gl}(n)$ was formed with a vector space of matrices and a commutator. It turns out it is isomorphic to Lie($GL(n)$). Can you find another vector space with a clever definition of a bracket such that it forms a Lie Algebra that is also isomorphic to Lie($G$)? Can you find all of them? or at least "describe" how would such a set be like? – crimson_sprite Oct 10 '17 at 10:08
• @Sebgr Well, you can't find all of them, because if $X$ is an uncountable set then it can be equiped with a Lie Algebra structure isomorphic to any $\mbox{Lie}(G)$ via any bijection (Lie Algebras of Lie Groups are of continuum cardinality). And since all sets equinumerous to $X$ form a proper class then there's not much you can do about it. – freakish Oct 10 '17 at 10:20
• Possibly "isomorphism" is an obsolete word for "injective homomorphism", and the question is about describing faithful representations, which is still somewhat vague. – YCor Oct 10 '17 at 16:39

The "variety" of all Lie algebra structures $\mathcal{L}_n(K)$ over a given vector space $V$ of dimension $n$ has points $(c_{ij}^k)\in K^{n^3}$, satisfying skew-symmetry and the Jacobi identity. If $(e_1,\ldots e_n)$ is a basis of $V$, then $[e_i,e_j]=\sum_{k=1}^n c_{ij}^ke_k$. So a fixed Lie algebra $L$ of dimension $n$ is given there by all sets of structure constants defining a Lie algebra isomorphic to $L$. The group $GL_n(K)$ acts on $\mathcal{L}_n(K)$ by base change. In this sense, one could say that "all possible isomorphic Lie algebras to the canonical one" are given by such sets of structure constants.