Lie algebras

DEFINITION 1.1   Let $K$ be a commutative ring. Then a Lie algebra $\mathfrak{g}$ over $K$ is a $K$ -module with a bilinear (non associative) bracket product (``Lie bracket'')

$$[\bullet,\bullet]: \mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$$

which satisfies the following axioms:

1. $[X,X]=0$ for all $X\in \mathfrak{g}$ .
2. (``Jacobi identity'')
   $$[X,[Y,Z]] =[[X,Y],Z]]+[Y,[X,Z]] \qquad (\forall X,Y,Z\in \mathfrak{g}).$$

EXAMPLE 1.2   Any associative algebra $A$ over $k$ may be regarded as a Lie algebra with the ``commutator'' as a Lie bracket.

In this talk, we always regard associative algebra as a Lie algebra equipped with the commutator product unless otherwise specified.

LEMMA 1.3   Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ . Then there exists an associative unital algebra $U(\mathfrak{g})$ with a Lie algebra homomorphism

$$\iota_\mathfrak{g}:\mathfrak{g}\to U(\mathfrak{g})$$

with the following universal property:

For any associative unital algebra $A$ with a Lie algebra homomorphism $\phi: \mathfrak{g}\to A$ , there exists a unique algebra homomorphism

$$\psi: U(\mathfrak{g})\to A$$

such that $\psi\circ \iota_\mathfrak{g}=\phi$ holds.

The pair $(U(\mathfrak{g}),\iota_\mathfrak{g})$ is unique up to an isomorphism.

DEFINITION 1.4   Under the assumption of the previous Lemma, The pair $(U(\mathfrak{g}),\iota_\mathfrak{g})$ is called the universal enveloping algebra the Lie algebra $\mathfrak{g}$ .

Universal enveloping algebras of Lie algebras form an important class of non commutative associative algebras. Our task in this Part is to describe these algebras in our language.