nLab
LieAlg

Definition

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

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

$\mathrm{Lie}\mathrm{Alg}$ 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 ${\wedge }^{•}𝔤$.

Special objects

• abelian Lie algebra

• simple Lie algebra

• semisimple Lie algebra

category: category