Homogeneous involutions on upper triangular matrices
Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees t...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Castilho de Mello, Thiago [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2022 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Nature Switzerland AG 2022 |
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| Übergeordnetes Werk: |
Enthalten in: Archiv der Mathematik - Springer International Publishing, 1948, 118(2022), 4 vom: 10. März, Seite 365-374 |
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| Übergeordnetes Werk: |
volume:118 ; year:2022 ; number:4 ; day:10 ; month:03 ; pages:365-374 |
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| DOI / URN: |
10.1007/s00013-022-01712-6 |
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OLC2078390631 |
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| 520 | |a Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. | ||
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| 650 | 4 | |a Graded algebras | |
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| 650 | 4 | |a Degree-inverting involutions | |
| 650 | 4 | |a Homogeneous involutions | |
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10.1007/s00013-022-01712-6 doi (DE-627)OLC2078390631 (DE-He213)s00013-022-01712-6-p DE-627 ger DE-627 rakwb eng 510 050 VZ 17,1 ssgn 31.00 bkl Castilho de Mello, Thiago verfasserin (orcid)0000-0002-6988-0910 aut Homogeneous involutions on upper triangular matrices 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2022 Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. Upper triangular matrices Graded algebras Involutions Graded involutions Degree-inverting involutions Homogeneous involutions Enthalten in Archiv der Mathematik Springer International Publishing, 1948 118(2022), 4 vom: 10. März, Seite 365-374 (DE-627)129061581 (DE-600)475-3 (DE-576)014392364 0003-889X nnns volume:118 year:2022 number:4 day:10 month:03 pages:365-374 https://doi.org/10.1007/s00013-022-01712-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4277 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 118 2022 4 10 03 365-374 |
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10.1007/s00013-022-01712-6 doi (DE-627)OLC2078390631 (DE-He213)s00013-022-01712-6-p DE-627 ger DE-627 rakwb eng 510 050 VZ 17,1 ssgn 31.00 bkl Castilho de Mello, Thiago verfasserin (orcid)0000-0002-6988-0910 aut Homogeneous involutions on upper triangular matrices 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2022 Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. Upper triangular matrices Graded algebras Involutions Graded involutions Degree-inverting involutions Homogeneous involutions Enthalten in Archiv der Mathematik Springer International Publishing, 1948 118(2022), 4 vom: 10. März, Seite 365-374 (DE-627)129061581 (DE-600)475-3 (DE-576)014392364 0003-889X nnns volume:118 year:2022 number:4 day:10 month:03 pages:365-374 https://doi.org/10.1007/s00013-022-01712-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4277 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 118 2022 4 10 03 365-374 |
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10.1007/s00013-022-01712-6 doi (DE-627)OLC2078390631 (DE-He213)s00013-022-01712-6-p DE-627 ger DE-627 rakwb eng 510 050 VZ 17,1 ssgn 31.00 bkl Castilho de Mello, Thiago verfasserin (orcid)0000-0002-6988-0910 aut Homogeneous involutions on upper triangular matrices 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2022 Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. Upper triangular matrices Graded algebras Involutions Graded involutions Degree-inverting involutions Homogeneous involutions Enthalten in Archiv der Mathematik Springer International Publishing, 1948 118(2022), 4 vom: 10. März, Seite 365-374 (DE-627)129061581 (DE-600)475-3 (DE-576)014392364 0003-889X nnns volume:118 year:2022 number:4 day:10 month:03 pages:365-374 https://doi.org/10.1007/s00013-022-01712-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4277 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 118 2022 4 10 03 365-374 |
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10.1007/s00013-022-01712-6 doi (DE-627)OLC2078390631 (DE-He213)s00013-022-01712-6-p DE-627 ger DE-627 rakwb eng 510 050 VZ 17,1 ssgn 31.00 bkl Castilho de Mello, Thiago verfasserin (orcid)0000-0002-6988-0910 aut Homogeneous involutions on upper triangular matrices 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2022 Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. Upper triangular matrices Graded algebras Involutions Graded involutions Degree-inverting involutions Homogeneous involutions Enthalten in Archiv der Mathematik Springer International Publishing, 1948 118(2022), 4 vom: 10. März, Seite 365-374 (DE-627)129061581 (DE-600)475-3 (DE-576)014392364 0003-889X nnns volume:118 year:2022 number:4 day:10 month:03 pages:365-374 https://doi.org/10.1007/s00013-022-01712-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4277 31.00 Mathematik: Allgemeines Mathematik: Allgemeines VZ AR 118 2022 4 10 03 365-374 |
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Homogeneous involutions on upper triangular matrices |
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Castilho de Mello, Thiago |
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Archiv der Mathematik |
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Archiv der Mathematik |
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eng |
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500 - Science 000 - Computer science, information & general works |
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2022 |
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Castilho de Mello, Thiago |
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118 |
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Castilho de Mello, Thiago |
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10.1007/s00013-022-01712-6 |
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510 050 |
| title_sort |
homogeneous involutions on upper triangular matrices |
| title_auth |
Homogeneous involutions on upper triangular matrices |
| abstract |
Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. © Springer Nature Switzerland AG 2022 |
| abstractGer |
Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. © Springer Nature Switzerland AG 2022 |
| abstract_unstemmed |
Abstract Let K be a field of characteristic different from 2 and let G be a group. If the algebra $$UT_n$$ of $$n\times n$$ upper triangular matrices over K is endowed with a G-grading $$\Gamma : UT_n=\oplus _{g\in G}A_g$$, we give necessary and sufficient conditions on $$\Gamma $$ that guarantees the existence of a homogeneous antiautomorphism on A, i.e., an antiautomorphism $$\varphi $$ satisfying $$\varphi (A_g)=A_{\theta (g)}$$ for some permutation $$\theta $$ of the support of the grading. It turns out that $$UT_n$$ admits a homogeneous antiautomorphism if and only if the reflection involution of $$UT_n$$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $$UT_n$$ is defined by the map $$\theta $$, then any other homogeneous antiautomorphism is defined by the same map $$\theta $$. © Springer Nature Switzerland AG 2022 |
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| title_short |
Homogeneous involutions on upper triangular matrices |
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https://doi.org/10.1007/s00013-022-01712-6 |
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