Root groupoid and related Lie superalgebras
Gorelik, Maria1; Hinich, Vladimir2; Serganova, Vera3
1 Department of Mathematics, Weizmann Institute of science POB 26 76100 Rehovot Israel
2 Department of Mathematics, University of Haifa Abba Khoushy Ave 199 Mt Carmel 3498838 Haifa Israel
3 Department of Mathematics, University of California 970 Evans Hall Berkeley, CA 94720-3840 United States
Annals of Representation Theory, Volume 1 (2024) no. 4, pp. 465-516.

We introduce a notion of a root groupoid as a replacement of the notion of a Weyl group for (Kac–Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an abstract groupoid the root groupoid has many connected components and we show that to some of them one can associate an interesting family of Lie superalgebras which we call root superalgebras. We classify root superalgebras satisfying some additional assumptions. To each root groupoid component we associate a graph (called the skeleton) generalizing the Cayley graph of the Weyl group. We establish the Coxeter property of the skeleton generalizing in this way the fact that the Weyl group of a Kac–Moody Lie algebra is Coxeter.

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DOI: 10.5802/art.13
Classification: 17B22, 17B67, 17B99
Keywords: Kac–Moody Lie superalgebras
Gorelik, Maria 1; Hinich, Vladimir 2; Serganova, Vera 3

1 Department of Mathematics, Weizmann Institute of science POB 26 76100 Rehovot Israel
2 Department of Mathematics, University of Haifa Abba Khoushy Ave 199 Mt Carmel 3498838 Haifa Israel
3 Department of Mathematics, University of California 970 Evans Hall Berkeley, CA 94720-3840 United States
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Gorelik, Maria; Hinich, Vladimir; Serganova, Vera. Root groupoid and related Lie superalgebras. Annals of Representation Theory, Volume 1 (2024) no. 4, pp. 465-516. doi : 10.5802/art.13. https://art.centre-mersenne.org/articles/10.5802/art.13/

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