Definition of Lie algebra presentations

Question

A group presentation is defined as a collection of generators (such that every element is a product of powers of those) and relations between the generators such that the group is uniquely defined by those generators and their relations.

My questions addresses the proper definition of presentations of Lie algebras. Are they defined completely analoguously such that we have

• A set of generators $X_{i}$ (such that every element is a linear combination of the $X_i$ and can be constructed from the Lie bracket of the $X_i$?)
• The Lie bracket relations between the $X_i$?

Would this suffice as a definition of a Lie algebra presentation or would there be missing something?

Answer 1

That's correct, but I would make what the $X_i$ are more clear. A Lie algebra is a vector space $V$, with Lie bracket $[\cdot, \cdot]: V \times V \to V$, so I would say a presentation is

1. A basis $\{X_i\}$ of $V$; along with
2. A collection of scalars $a_{ij}^k$ such that $[X_i, X_j] = \sum_k a_{ij}^k X_k$.

Which informally, is just some basis $\{X_i\}$ of $V$ that you know the Lie bracket on. The scalars $a_{ij}^k$ are called the structure constants of the Lie algebra, and they must satsify certain relations, due to the fact that they come from a Lie bracket, not any scalars will do. For example, we must have $a_{ii}^k = 0$, and $a_{ij}^k = -a_{ji}^k$, amongst others.

There is a different notion of just bare generators and relations, which is closer to a group presentation. But, like group presentations, they can be quite difficult to work with, and you might end up having to send some of your generators $X_i$ to $0$ in order to satisfy the relations.