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Article Dans Une Revue Annali della Scuola Normale Superiore di Pisa, Classe di Scienze Année : 2021

Homology of the Lie algebra gl(∞, R)

Résumé

In this note we compute the homology of the Lie algebra gl(∞, R) where R is an associative unital k-algebra which is used in higher dimensional soliton theory [Ta]. When k is a field of characteristic 0, our result justifies an old result of Feigin and Tsygan [FT]. The special case when R = k = C appeared first in soliton theory (cf. [JM]). Dedicated to the memory of our friend Jean-Louis Loday 0. Introduction Among several versions of the Lie algebra gl of infinite rank, the Lie algebra gl(∞), that has been extensively used to describe the soliton solutions of the Kadomtsev-Petviashvili (KP in short) hierarchy (see, e.g., [DJM] for detail) in the first half of 1980's, has a special feature. For example, the Lie algebra gl(∞) is neither ind-finite nor pro-finite. For this reason, it had been a difficult task to analyze its algebraic properties. In 1983, B. Feigin and B. Tsygan published a short note [FT] (only 2 pages long !) where they determined the homology of the Lie algebra gl(∞, k) where k is a field of characteristic 0. They denoted this Lie algebra by gJ(k) that is recalled in §1.1. Unfortunately, it seems that their paper is too dense to decompress, so that this article had not been studied carefully in the mathematical community. At the same time, their note generated much interest, and-even 34 years later-the statements are important. In this paper, we managed to compute the homology of the Lie algebra gl(∞, R), where R is an associative unital k-algebra and k is a field of characteristic 0. With our proofs the special case when R = k also turns out to become clear. That is the case Feigin and Tsygan discussed briefly in their note [FT]. We hope that our paper-beside generalizing the case which shows up in soliton theory-also makes the article [FT] accessible to the mathematical community. One of the amazing facts we show is The primitive part of H • (gJ(R)) = HC • (R)[2] = (The primitive part of H • (gl ∞ (R)))[1], that is, gl ∞ (R) is obtained by delooping gJ(R) at the homological level ! Technically, two big problems had been already solved: the Loday-Quillen-Tsygan theorem (cf. Theorem 2.1) by J. L. Loday and D. Quillen [LQ] and B. Tsygan [Ts], and a generalization of the Hochschild-Serre spectral sequence (cf. Theorem 3.1) by D. Stefan [St]. Thus, the main task for us is to combine these facts in a suitable manner and compute explicitly a specific spectral sequence. Throughout this paper, a field k of characteristic 0 is fixed.
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Dates et versions

hal-01665476 , version 1 (15-12-2017)

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Citer

A. Fialowski, K. Iohara. Homology of the Lie algebra gl(∞, R). Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2021, Homotopy Algebras, Deformation Theory and Quantization, 22 (2), pp.835 -- 852. ⟨10.2422/2036-2145.201902_012⟩. ⟨hal-01665476⟩
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