Universal enveloping algebra ($U(\mathfrak{g})$)

The universal enveloping algebra $U(\mathfrak{g})$ is the associative algebra generated by $\mathfrak{g}$ subject to $xy-yx=[x,y]$. Its modules are the same as Lie algebra representations.

For a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, the shifted Harish-Chandra map identifies the centre of the universal enveloping algebra with the Weyl group-invariant polynomial functions on ${\mathfrak{h}}^{\ast }$. To normalize the shift, let $M^{\chi }$ have highest weight $\chi -\rho$, where $\rho$ is the Weyl vector. A central element $z$ acts by a scalar $j_z(\chi )$, and $z\mapsto j_z$ is the displayed algebra isomorphism. The Poincaré-Birkhoff-Witt theorem gives polynomial dependence on $\chi$ by projecting $z$ onto its pure Cartan component and evaluating at $\chi -\rho$. The associated central characters agree exactly when their shifted parameters lie in the same Weyl orbit. Without incorporating the shift into the parameter, the invariance is expressed using the dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$.

For an ordered basis $(x_i)$ of a Lie algebra $\mathfrak{g}$, the ordered monomials $x_1^{a_1}\cdots x_n^{a_n}$ form a basis of $U(\mathfrak{g})$. In particular, a triangular decomposition $\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{t}\oplus {\mathfrak{n}}^{+}$ gives $U(\mathfrak{g})\cong U({\mathfrak{n}}^{-})\otimes U(\mathfrak{t})\otimes U({\mathfrak{n}}^{+})$ as vector spaces.

For a graded Lie algebra over characteristic zero, the associated graded algebra of its universal enveloping algebra under the word-length filtration is its graded symmetric algebra. Ordered homogeneous monomials form a basis, with exponent at most one on each odd Lie basis element. The defining graded commutation relations reorder every word, and the Jacobi identity resolves the triple-reordering ambiguities. In particular the primitive elements of a Hopf algebra of $U(L)$ are precisely $L$: in its associated graded algebra, a primitive polynomial of symmetric length greater than one would have a nonzero mixed term under the diagonal, which is impossible in characteristic zero.