Poincaré–Birkhoff–Witt theorems

Summary

We sample some Poincaré–Birkhoff–Witt theorems appearing in mathematics. Along the way, we compare modern techniques used to establish such results, for example, the composition-diamond lemma, Gröbner basis theory, and the homological approaches of Braverman and Gaitsgory and of Polishchuk and Positselski. We discuss several contexts for PBW theorems and their applications, such as Drinfeld–Jimbo quantum groups, graded Hecke algebras, and symplectic reflection and related algebras.

Poincaré [1900] published a fundamental result on Lie algebras that would prove a powerful tool in representation theory: A Lie algebra embeds into an associative algebra that behaves in many ways like a polynomial ring. Capelli [1890] had proven a special case of this theorem, for the general linear Lie algebra, ten years earlier. Birkhoff [1937] and Witt [1937] independently formulated and proved versions of the theorem that we use today, although neither author cited this earlier work. The result was called the Birkhoff–Witt theorem for years and then later the Poincaré–Witt theorem (see [Cartan and Eilenberg 1956]) before Bourbaki [1960] prompted use of its current name, the Poincaré–Birkhoff–Witt theorem.

The original theorem on Lie algebras was greatly expanded over time by a number of authors to describe various algebras, especially those defined by quadratictype relations (including Koszul rings over semisimple algebras). Poincaré– Birkhoff–Witt theorems are often used as a springboard for investigating the representation theory of algebras. These theorems are used to

• reveal an algebra as a deformation of another, well-behaved algebra,

• posit a convenient basis (of “monomials”) for an algebra, and

• endow an algebra with a canonical homogeneous (or graded) version.

In this survey, we sample some of the various Poincaré–Birkhoff–Witt theorems, applications, and techniques used to date for proving these results. Our survey is not intended to be all-inclusive; we instead seek to highlight a few of the more recent contributions and provide a helpful resource for users of Poincaré– Birkhoff–Witt theorems, which we henceforth refer to as PBW theorems.