An associative left brace is a ring. (English) Zbl 1467.16037
To approach involutive set-theoretic solutions of the Yang-Baxter equation [W. Rump, J. Algebra 307, 153–170 (2007; Zbl 1115.16022)] proposed to consider (one sided) braces - algebraic structures which generalize radical rings.
An algebra \((B,+,\cdot)\) is called a left brace if \((B,+)\) is an abelian group, \((B,\cdot)\) is a group and the operations satisfy for all \(a,b,c\in B\) the condition: \(a\cdot b+a\cdot c = a\cdot(b+c)+a\). A right brace is defined analogously by: \(a\cdot c+b\cdot c=(a+b)\cdot c+c\). A left brace which is also a right one is called two-sided brace.
In any left brace \((B,+,\cdot)\) the circle binary operation \(\circ\) defined for \(a,b\in B\) by: \(a\circ b=a\cdot b-a-b\) is left distributive with respect to the addition \(+\). But in general, the circle operation is neither right distributive with respect to the addition \(+\) nor associative.
It is well known that a left brace \((B,+,\cdot)\) is two sided brace if and only if the circle operation \(\circ\) is right distributive with respect to the addition \(+\) or equivalently if and only if \((B,+,\circ)\) is a Jacobson radical ring [loc. cit.].
Cedó, Gateva-Ivanova and Smoktunowicz stated in [F. Cedó et al., J. Pure Appl. Algebra 222, No. 12, 3877–3890 (2018; Zbl 1427.16027)] the following question (Question 2.1 (2)): ”Does it exist a left brace \((B,+,\cdot)\), such that \((B,\circ)\) is a semigroup, but \((B,+,\cdot)\) is not a two-sided brace?”. At the same time they show (Proposition 2.2) that any left brace \((B,+,\cdot)\) with the additive group \((B,+)\) without elements of order \(2\) and associative the circle operation is two-sided.
The author of this paper extends the result and proves that arbitrary left brace \((B,+,\cdot)\) with associative the circle operation is two-sided (Theorem 1.1). In consequence, in this case \((B,+,\circ)\) is a Jacobson radical ring.
An algebra \((B,+,\cdot)\) is called a left brace if \((B,+)\) is an abelian group, \((B,\cdot)\) is a group and the operations satisfy for all \(a,b,c\in B\) the condition: \(a\cdot b+a\cdot c = a\cdot(b+c)+a\). A right brace is defined analogously by: \(a\cdot c+b\cdot c=(a+b)\cdot c+c\). A left brace which is also a right one is called two-sided brace.
In any left brace \((B,+,\cdot)\) the circle binary operation \(\circ\) defined for \(a,b\in B\) by: \(a\circ b=a\cdot b-a-b\) is left distributive with respect to the addition \(+\). But in general, the circle operation is neither right distributive with respect to the addition \(+\) nor associative.
It is well known that a left brace \((B,+,\cdot)\) is two sided brace if and only if the circle operation \(\circ\) is right distributive with respect to the addition \(+\) or equivalently if and only if \((B,+,\circ)\) is a Jacobson radical ring [loc. cit.].
Cedó, Gateva-Ivanova and Smoktunowicz stated in [F. Cedó et al., J. Pure Appl. Algebra 222, No. 12, 3877–3890 (2018; Zbl 1427.16027)] the following question (Question 2.1 (2)): ”Does it exist a left brace \((B,+,\cdot)\), such that \((B,\circ)\) is a semigroup, but \((B,+,\cdot)\) is not a two-sided brace?”. At the same time they show (Proposition 2.2) that any left brace \((B,+,\cdot)\) with the additive group \((B,+)\) without elements of order \(2\) and associative the circle operation is two-sided.
The author of this paper extends the result and proves that arbitrary left brace \((B,+,\cdot)\) with associative the circle operation is two-sided (Theorem 1.1). In consequence, in this case \((B,+,\circ)\) is a Jacobson radical ring.
Reviewer: Agata Pilitowska (Warszawa)
MSC:
| 16T25 | Yang-Baxter equations |
| 16N20 | Jacobson radical, quasimultiplication |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
References:
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| [2] | Cedó, F., Left braces: Solutions of the Yang-Baxter equation, Adv. Group Theory Appl.5 (2018) 33-90. https://doi.org/10.4399/97888255161422 · Zbl 1403.16033 |
| [3] | Cedó, F., Gateva-Ivanova, T. and Smoktunowicz, A., Braces and symmetric groups with special conditions, J. Pure Appl. Algebra222(12) (2018) 3877-3890. https://doi.org/10.1016/j.jpaa.2018.02.012 · Zbl 1427.16027 |
| [4] | Cedó, F., Jespers, E. and Okniński, J., Braces and the Yang-Baxter equation, Comm. Math. Phys.327(1) (2014) 101-116. https://doi.org/10.1007/s00220-014-1935-y · Zbl 1287.81062 |
| [5] | Guarnieri, L. and Vendramin, L., Skew braces and the Yang-Baxter equation, Math. Comp.86(307) (2017) 2519-2534. https://doi.org/10.1090/mcom/3161 · Zbl 1371.16037 |
| [6] | Konovalov, A., Smoktunowicz, A. and Vendramin, L., On skew braces and their ideals, Exp. Math. (2018) 1-10. https://doi.org/10.1080/10586458.2018.1492476 |
| [7] | Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra307(1) (2007) 153-170. https://doi.org/10.1016/j.jalgebra.2006.03.040 · Zbl 1115.16022 |
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