<h4>1.1 <span class="Heading">Definition and examples</span></h4>
<p>A skew brace is a triple <span class="Math">(A,+,\circ)</span>, where <span class="Math">(A,+)</span> and <span class="Math">(A,\circ)</span> are two (not necessarily abelian) groups such that the compatibility <span class="Math">a\circ (b+c)=a\circ b-a+a\circ c</span> holds for all <span class="Math">a,b,c\in A</span>. Ones proves that the map <span class="Math">\lambda\colon (A,\circ)\to\mathrm{Aut}(A,+)</span>, <span class="Math">a\mapsto\lambda_a(b)</span>, <span class="Math">\lambda_a(b)=-a+a\circ b</span>, is a group homomorphism. Notation: For <span class="Math">a,b\in A</span>, we write <span class="Math">a*b=\lambda_a(b)-b</span>.</p>
<p>The argument <var class="Arg">list</var> is a list of pairs of elements in a group. By Proposition 5.11 of <a href="chapBib.html#biBMR3647970">[GV17]</a>, skew braces over an abelian group <span class="Math">A</span> are equivalent to pairs <span class="Math">(G,\pi)</span>, where <span class="Math">G</span> is a group and <span class="Math">\pi\colon G\to A</span> is a bijective <span class="Math">1</span>-cocycle, a finite skew brace can be constructed from the set <span class="Math">\{(a_j,g_j):1\leq j\leq n\}</span>, where <span class="Math">G=\{g_1,\dots,g_n\}</span> and <span class="Math">A=\{a_1,\dots,a_n\}</span> are permutation groups. This function is used to construct skew braces.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Skewbrace([[(),()]]);</span>
<brace of size 1>
<span class="GAPprompt">gap></span> <span class="GAPinput">Skewbrace([[(),()],[(1,2),(1,2)]]);</span>
<brace of size 2>
</pre></div>
<p>This function returns the trivial brace over the abelian group <var class="Arg">abelian_group</var>. Here <var class="Arg">abelian_group</var> should be an abelian group!</p>
<p>The function returns <var class="Arg">[ n, k ]</var> if the skew brace <var class="Arg">obj</var> is isomorphic to <var class="Arg">SmallSkewbrace(n,k)</var>.</p>
<p>The function returns <var class="Arg">[ n, k ]</var> if the brace of abelian type <var class="Arg">obj</var> is isomorphic to <var class="Arg">SmallBrace(n,k)</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismSkewbraces</code>( <var class="Arg">obj1</var>, <var class="Arg">obj2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an isomorphism of skew braces if <var class="Arg">obj1</var> and <var class="Arg">obj2</var> are isomorphic and <var class="Arg">fail</var> otherwise.</p>
<p>If <span class="Math">A</span> and <span class="Math">B</span> are skew braces, a skew brace homomorphism is a map <span class="Math">f\colon A\to B</span> such that</p>
<p>hold for all <span class="Math">a,b\in A</span>. A skew brace isomorphism is a bijective skew brace homomorphism. <var class="Arg">IsomorphismSkewbraces</var> first computes all injective homomorphisms from <span class="Math">(A,+)</span> to <span class="Math">(B,+)</span> and then tries to find one <span class="Math">f</span> such that <span class="Math">f(a\circ b)=f(a)\circ f(b)</span> for all <span class="Math">a,b\in A</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTwoSided</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace is two sided, <var class="Arg">false</var> otherwise</p>
<p>A skew brace <span class="Math">A</span> is said to be <em>two-sided</em> if <span class="Math">(a+b)\circ c=a\circ c-c+b\circ c</span> holds for all <span class="Math">a,b,c\in A</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAutomorphismGroupOfSkewbrace</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the group is the automorphism group of a skew braces, <varclass="Arg">false</var> otherwise</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsClassical</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace is of abelian type, <var class="Arg">false</var> otherwise</p>
<p>Let <span class="Math">\mathcal{X}</span> be a property of groups. A skew brace <span class="Math">A</span> is said to be of <span class="Math">\mathcal{X}</span>-type if its additive group belongs to <span class="Math">\mathcal{X}</span>. In particular, skew braces of abelian type are those skew braces with abelian additive group. Such skew braces were introduced by Rump in <a href="chapBib.html#biBMR2278047">[Rum07]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOfNilpotentType</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace is of nilpotent type, <var class="Arg">false</var> otherwise</p>
<p>Let <span class="Math">\mathcal{X}</span> be a property of groups. A skew brace <span class="Math">A</span> is said to be of <span class="Math">\mathcal{X}</span>-type if its additive group belongs to <span class="Math">\mathcal{X}</span>. In particular, skew braces of nilpotent type are those skew braces with nilpotent additive group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTrivialSkewbrace</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the skew brace is trivial, <var class="Arg">false</var> otherwise</p>
<p>The function returns <var class="Arg">true</var> if the skew brace <span class="Math">A</span> is trivial, i.e., <span class="Math">a\circ b=a+b</span> for all <span class="Math">a,b\in A</span>. WARNING: The property IsTrivial applied to a skew brace will return true if and only if the skew brace has only one element.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Skewbrace2YB</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the set-theoretic solution associated with the skew brace <var class="Arg">obj</var></p>
<p>If <span class="Math">A</span> is a skew brace, the map <span class="Math">r_A\colon A\times A\to A\times A</span></p>
<p class="pcenter">r_A(a,b)=(\lambda_a(b),\lambda_a(b)'\circ a\circ b)
<p>is a non-degenerate set-theoretic solution of the Yang--Baxter equation. Furthermore, <span class="Math">r_A</span> is involutive if and only if <span class="Math">A</span> is of abelian type (i.e., the additive group of <span class="Math">A</span> is abelian).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SkewbraceSubset2YB</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the set-theoretic solution associated with a given subset of a skew brace</p>
<p>Let <span class="Math">A</span> and <span class="Math">B</span> be two skew braces and <span class="Math">\sigma</span> be a skew brace action of <span class="Math">B</span> on <span class="Math">A</span>, this is a group homomorphism <span class="Math">\sigma\colon (B,\circ)\to Aut_{\mathrm{Br}}(A)</span> from the multiplicative group of <span class="Math">B</span> to the skew brace automorphism of <span class="Math">A</span>. The semidirect product of <span class="Math">A</span> and <spanclass="Math">B</span> with with respect to <span class="Math">\sigma</span> is the skew brace <span class="Math">A\rtimes_{\sigma}B</span> with operations</p>
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