Let E(n) be the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$2^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-dimensional Hopf algebra generated by anti-commuting elements <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$g,x_1, \ldots , x_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>,</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, with g grouplike, each <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$x_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> skew-primitive, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$g^2=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>g</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$x_i^2=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article we prove that E(n)-coactions over a finite-dimensional algebra A are classified by tuples <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$(\varphi , d_1, \ldots , d_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> consisting of an involution <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> and a family <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$(d_i)_{i=1,\ldots ,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>-derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1399_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$(c, u_1, \ldots , u_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, whenever A is a semisimple Clifford algebra.

E(n)-coactions on Semisimple Clifford Algebras

期刊论文
发表日期：

2025年7月10日

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Fabio Renda

摘要

Let E(n) be the $2^{n+1}$ $2^{n + 1}$ -dimensional Hopf algebra generated by anti-commuting elements $g,x_1, \ldots , x_n$ $g, x_{1}, \dots, x_{n}$ , with g grouplike, each $x_i$ $x_{i}$ skew-primitive, and $g^2=1$ $g^{2} = 1$ , $x_i^2=0$ $x_{i}^{2} = 0$ . In this article we prove that E(n)-coactions over a finite-dimensional algebra A are classified by tuples $(\varphi , d_1, \ldots , d_n)$ $(φ, d_{1}, \dots, d_{n})$ consisting of an involution $\varphi $ $φ$ and a family $(d_i)_{i=1,\ldots ,n}$ ${(d_{i})}_{i = 1, \dots, n}$ of $\varphi $ $φ$ -derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements $(c, u_1, \ldots , u_n)$ $(c, u_{1}, \dots, u_{n})$ , whenever A is a semisimple Clifford algebra.