Connections between properties of the additive and the multiplicative groups of a two-sided skew brace

Nasybullov, Timur

doi:10.1016/j.jalgebra.2019.05.005

Cited by 21 publications

(15 citation statements)

References 12 publications

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“…The following example gives a switch on a skew brace (see, for example, [22,35] for details about skew braces).…”

Section: 1mentioning

confidence: 99%

Multi-switches and representations of braid groups

Bardakov¹,

Nasybullov²

2019

Preprint

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In the paper we introduce the notion of a (virtual) multi-switch which generalizes the notion of a (virtual) switch. Using (virtual) multi-switches we introduce a general approach how to construct representations of (virtual) braid groups by automorphisms of algebraic systems. As a corollary we introduce new representations of virtual braid groups which generalize several previously known representations.

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“…The following example gives a switch on a skew brace (see, for example, [22,35] for details about skew braces).…”

Section: 1mentioning

confidence: 99%

Multi-switches and representations of braid groups

Bardakov¹,

Nasybullov²

2019

Preprint

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“…Skew braces have connections with other algebraic structures such as groups with exact factorizations, Zappa-Szép products, triply factorized groups and Hopf-Galois extensions [18]. Some algebraic aspects of skew braces are studied in [6,7,10,14,16,18]. Thanks to its relations with the Yang-Baxter equation skew braces can be applied for constructing representations of virtual braid groups and invariants of virtual links [2,3].…”

Section: Introductionmentioning

confidence: 99%

“…The first problem from the list above was studied, for example, in [5,16,18,20]. The most general result about this question is obtained in [20], where it is proved that the answer to this question is negative in the case of finite skew braces.…”

Section: Introductionmentioning

confidence: 99%

Finite skew braces with solvable additive group

Gorshkov¹,

Nasybullov²

2020

Preprint

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A. Smoktunowicz and L. Vendramin conjectured that if A is a finite skew brace with solvable additive group, then the multiplicative group of A is solvable. In this short note we make a step towards positive solution of this conjecture proving that if A is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group of A is not simple. On the way to obtaining this result, we prove that the conjecture of A. Smoktunowicz and L. Vendramin is correct in the case when the order of A is not divisible by 3.

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“…Besides their importance in constructing invariants for virtual knots and links, biquandles can be used for constructing representations of (virtual) braid groups [5,23] and solutions of the well-known Yang-Baxter equation [12] which has wide applications beyond mathematics (for other algebraic systems which give solutions of the Yang-Baxter equation, see, for example, [29,44,50]).…”

Section: Introductionmentioning

confidence: 99%

General constructions of biquandles and their symmetries

Bardakov¹,

Nasybullov²,

Singh³

2019

Preprint

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Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.

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Connections between properties of the additive and the multiplicative groups of a two-sided skew brace

Cited by 21 publications

References 12 publications

Multi-switches and representations of braid groups

Multi-switches and representations of braid groups

Finite skew braces with solvable additive group

General constructions of biquandles and their symmetries

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