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

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 \(A\times A\to V\).

MSC:

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