On regular subgroups of the affine group. (English) Zbl 1314.20001

Catino, Francesco; Colazzo, Ilaria; Stefanelli, Paola

Bull. Aust. Math. Soc. 91, No. 1, 76-85 (2015).

Let \(F\) be a field. Regular subgroups \(G\) of the affine group of an \(F\)-vector space are known to be equivalent to \(F\)-braces [F. Catino and R. Rizzo, Bull. Aust. Math. Soc. 79, No. 1, 103-107 (2009; Zbl 1184.20001)]. If \(G\) is abelian, an \(F\)-brace is the same as a radical \(F\)-algebra [A. Caranti et al., Publ. Math. 69, No. 3, 297-308 (2006; Zbl 1123.20002)]. \(F\)-Braces can be regarded as bijective 1-cocycles of an \(F\)-linear group representation, and they embrace other structures like groups of I-type, binomial skew-polynomial rings, and unitary set-theoretic solutions of the Yang-Baxter equation.
In the paper under review, the authors use the term “\(F\)-brace” in a more general sense. Inspired by their relationship to radical rings, they speak of “radical \(F\)-braces”, so that the more general concept comes close to \(F\)-algebras. For generalized \(F\)-braces, Hochschild 2-cocycles are introduced, and it is shown that any (radical) \(F\)-brace \(B\) with non-zero annihilator \(V\) can be reconstructed from \(A:=F/V\) by means of a 2-cocycle \(A\times A\to V\).

Reviewer: Wolfgang Rump (Stuttgart)

MSC:

20B10	Characterization theorems for permutation groups
16N20	Jacobson radical, quasimultiplication
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

affine groups; regular subgroups; braces; Yang-Baxter equation; radical rings

Citations:

Zbl 1184.20001; Zbl 1123.20002

References:

[1]	Rump, Note Mat. 34 pp 115– (2014)
[2]	Tamburini Bellani, Int. J. Group Theory 1 pp 17– (2012)
[3]	DOI: 10.1016/j.jalgebra.2006.03.040 · Zbl 1115.16022
[4]	DOI: 10.1007/978-1-4757-0163-0
[5]	DOI: 10.1016/j.jpaa.2012.06.012 · Zbl 1266.81112
[6]	Liebeck, Mem. Amer. Math. Soc. 203 (2009)
[7]	DOI: 10.1006/jabr.2000.8547 · Zbl 0972.20001
[8]	DOI: 10.1215/S0012-7094-47-01473-7 · Zbl 0029.34201
[9]	DOI: 10.1006/jabr.1999.8176 · Zbl 0953.20040
[10]	DOI: 10.1090/S0002-9947-2012-05503-6 · Zbl 1287.12002
[11]	DOI: 10.1007/s00220-014-1935-y · Zbl 1287.81062
[12]	DOI: 10.1017/S0004972708001068 · Zbl 1184.20001
[13]	Caranti, Publ. Math. Debrecen 69 pp 297– (2006)
[14]	Aczél, On Applications and Theory of Functional Equations (1969) · Zbl 0176.12801
[15]	DOI: 10.1142/S0219498808002904 · Zbl 1153.81505