nLab LieAlg

Definition

The category $Lie Alg$ is that whose objects are Lie algebras $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ and whose morphisms are Lie algebra homomorphisms, that is linear maps $\phi\colon \mathfrak{g} \to \mathfrak{h}$ such that for all $x,y \in \mathfrak{g}$ we have

$$\phi( [x,y]_{\mathfrak{g}}) = [\phi(x),\phi(y)]_\mathfrak{h} \,.$$

If Lie algebras are expressed in terms of their Chevalley–Eilenberg algebras (and if restricted to finite-dimensional Lie algebras), this may equivalently be characterized as follows:

$Lie Alg$ is the full subcategory of the opposite category of the category dgAlg of dg-algebras on those dg-algebras whose underlying graded algebra is a Grassmann algebra, i.e. of the form $\wedge^\bullet \mathfrak{g}$.

Special objects

• abelian Lie algebra

• simple Lie algebra

• semisimple Lie algebra

• LieGrp

• Lie's three theorems

category: category