Degree-one almost commutative algebra ($F_{0}A=k,A=⟨F_{1}A⟩,[F_{1}A,F_{1}A]\subseteq F_{1}A$)

This is the degree-one convention for an almost commutative algebra: the exhaustive filtration is generated by a finite-dimensional first step containing $1$, the degree-zero part is $k$, and its associated graded algebra is commutative. The first step is a finite-dimensional Lie algebra under commutators. Its inclusion into $A$ extends by the universal property to a surjection from its universal enveloping algebra. The element $1$ of the first step is retained as a central Lie generator and its image is the associative identity; quotienting the first step by $k1$ before taking the bracket can lose scalar commutators. This convention is stronger than merely requiring a finitely generated commutative associated graded algebra for an arbitrary weighted filtration.

// Destination: noncommutative-algebra.bigb