Involutive Yang-Baxter groups

Abstract

In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"><mml:semantics><mml:mi>I</mml:mi><mml:annotation encoding="application/x-tex">I</mml:annotation></mml:semantics></mml:math></inline-formula>-type. A group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {G}</mml:annotation></mml:semantics></mml:math></inline-formula>of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"><mml:semantics><mml:mi>I</mml:mi><mml:annotation encoding="application/x-tex">I</mml:annotation></mml:semantics></mml:math></inline-formula>-type is a group isomorphic to a subgroup of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper F normal a Subscript n Baseline right-normal-factor-semidirect-product normal upper S normal y normal m Subscript n"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">a</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo>⋊</mml:mo><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Fa}_n\rtimes \mathrm {Sym}_n</mml:annotation></mml:semantics></mml:math></inline-formula>so that the projection onto the first component is a bijective map, where<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper F normal a Subscript n"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">F</mml:mi><mml:mi mathvariant="normal">a</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\mathrm {Fa}_n</mml:annotation></mml:semantics></mml:math></inline-formula>is the free abelian group of rank<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal y normal m Subscript n"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\mathrm {Sym}_{n}</mml:annotation></mml:semantics></mml:math></inline-formula>is the symmetric group of degree<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>. The projection of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {G}</mml:annotation></mml:semantics></mml:math></inline-formula>onto the second component<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal y normal m Subscript n"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">m</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\mathrm {Sym}_n</mml:annotation></mml:semantics></mml:math></inline-formula>we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld’s problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>, classify the groups of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"><mml:semantics><mml:mi>I</mml:mi><mml:annotation encoding="application/x-tex">I</mml:annotation></mml:semantics></mml:math></inline-formula>-type with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"><mml:semantics><mml:mi>I</mml:mi><mml:annotation encoding="application/x-tex">I</mml:annotation></mml:semantics></mml:math></inline-formula>-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.