Universal enveloping superalgebra

Let $\mathfrak{g}$ be a complex Lie superalgebra. Give its tensor algebra $T(\mathfrak{g})={⨁}_{n\ge 0}{\mathfrak{g}}^{\otimes n}$ the supergrading in which a homogeneous tensor $x_1\otimes \cdots \otimes x_n$ has parity $\mathrm{\mid }{x}_1\mathrm{\mid }+\cdots +\mathrm{\mid }{x}_n\mathrm{\mid }(\mathrm{m}\mathrm{o}\mathrm{d}2)$, and use concatenation as multiplication (Tensor algebra of a vector space). Let $I$ be the two-sided ideal generated by the following homogeneous relations (two-sided ideal and generated ideal are as in Left, right and two-sided ideals and The ideal generated by a subset and principal ideals): $x\otimes y-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}y\otimes x-[x,y]$ for homogeneous $x,y\in \mathfrak{g}$. Each generator has parity $\mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid }$ because the bracket is graded. The lemma The ideal generated by homogeneous elements of a graded algebra is graded therefore makes $I$ graded, so the quotient has the induced supergrading. The quotient $U(\mathfrak{g}):=T(\mathfrak{g})\mathrm{/}I$ with the quotient multiplication is the universal enveloping superalgebra. Write ${\iota }_{\mathfrak{g}}:\mathfrak{g}\to U(\mathfrak{g})$ for the degree-one map. The defining relation is ${\iota }_{\mathfrak{g}}(x){\iota }_{\mathfrak{g}}(y)-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}{\iota }_{\mathfrak{g}}(y){\iota }_{\mathfrak{g}}(x)={\iota }_{\mathfrak{g}}([x,y])\mathrm{.}$ In particular, if $x$ is odd then $2{\iota }_{\mathfrak{g}}(x{)}^2={\iota }_{\mathfrak{g}}([x,x])$, hence ${\iota }_{\mathfrak{g}}(x{)}^2=\frac{1}{2}{\iota }_{\mathfrak{g}}([x,x])$ over $\mathbb{C}$. This quotient definition does not itself assert an ordered-monomial basis; the PBW basis is a separate theorem.