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Zbl 0596.17009
Neher, Erhard
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)
[J] Jahresber. Dtsch. Math.-Ver. 87, 164-181 (1985). ISSN 0012-0456; ISSN 1869-7135/e

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,${\bbfC})$. Finally he discusses the generalization to other affine Lie algebras which yield Macdonald's $\eta$-identities [{\it I. G. Macdonald}, Invent. Math. 15, 91-143 (1972; Zbl 0244.17005)], exemplified by $\eta\sp{24}$ and $\eta\sp{248}$. There is a bibliography of 61 items.
[G.Köhler]

MSC 2000:: *17B65 Infinite-dimensional Lie algebras
11F11 Modular forms, one variable
05A19 Combinatorial identities
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

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