= Degree-one almost commutative algebra
{title2=$F_0A=k,\quad A=\langle F_1A\rangle,\quad[F_1A,F_1A]\subseteq F_1A$}
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
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