PrimitiveLie

PrimitiveLie is not a standard term with a single universal definition in mathematics. In typical discussions within Lie theory and Hopf algebras, the phrase can refer to the Lie algebra formed by the primitive elements of a Hopf algebra, particularly the universal enveloping algebra of a Lie algebra.

An element x in a Hopf algebra H is called primitive if its coproduct satisfies Δ(x) = x ⊗ 1 + 1 ⊗ x. The set P(H) of all primitive elements is closed under the Lie bracket [x, y] = xy − yx, making P(H) into a Lie algebra. A central example occurs with the universal enveloping algebra U(g) of a Lie algebra g: the primitive elements of U(g) correspond to g, yielding a canonical identification P(U(g)) ≅ g. This connection shows how Lie algebra structure sits inside the broader Hopf-algebra framework.

In addition to theoretical usage, primitiveLie can also denote a software project or library focused on the study or computation of primitive elements and primitive Lie algebras. Such implementations typically provide data structures for algebras and coalgebras, routines for computing coproducts, counits, and brackets, and algorithms for extracting primitive elements or constructing Lie algebras from Hopf data. When used in this way, primitiveLie services aim to facilitate explicit computations and explorations of the relationship between Hopf algebra primitives and Lie algebras.

See also: Lie algebra, Hopf algebra, primitive element, universal enveloping algebra, primitive cohomology.