Jacobis Tripelprodukt-Identität und \(\eta\)-Identitäten in der Theorie affiner Lie-Algebren. (Jacobi’s triple product identity and \(\eta\)- identities in the theory of affine Lie algebras). (German) Zbl 0596.17009

Neher, Erhard

Jahresber. Dtsch. Math.-Ver. 87, 164-181 (1985).

This is a nicely readable written version of the author’s inaugural lecture. It begins with a report on classical results in partition theory: Euler’s pentagonal number theorem, Jacobi’s triple product identity, and identities for the first and third power of Dedekind’s \(\eta\)-function. Then the author explains the transcription of Jacobi’s triple product identity into a formula on the root system and the Weyl group of the affine Lie algebra attached to sl(2,\({\mathbb{C}})\). Finally he discusses the generalization to other affine Lie algebras which yield Macdonald’s \(\eta\)-identities [I. G. Macdonald, Invent. Math. 15, 91-143 (1972; Zbl 0244.17005)], exemplified by \(\eta^{24}\) and \(\eta^{248}\). There is a bibliography of 61 items.

Reviewer: G.Köhler

MSC:

17B65	Infinite-dimensional Lie (super)algebras
11F11	Holomorphic modular forms of integral weight
05A19	Combinatorial identities, bijective combinatorics
11P81	Elementary theory of partitions

Keywords:

Euler’s pentagonal number theorem; Jacobi’s triple product identity; Dedekind’s \(\eta \) -function; root system; affine Lie algebras; Macdonald’s \(\eta \) -identities; bibliography

Citations:

Zbl 0244.17005