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Alternating 3-Forms and Exceptional Simple Lie Groups of Type G 2
It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connect...
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Published in: | Canadian journal of mathematics 1983-10, Vol.35 (5), p.776-806 |
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container_title | Canadian journal of mathematics |
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creator | Herz, Carl |
description | It is now customary to give concrete descriptions of the exceptional simple Lie groups of type
G
2
as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let
W
be a complex 7-dimensional vector space. Among the alternating 3-forms on
W
there is a connected dense open subset Ψ(
W
) of “maximal” forms. If
ψ
∈ Ψ(
W
) then the subgroup of AUT
C
(
W
) consisting of the invertible complex-linear transformations
S
such that
ψ
(
S•, S•, S•
) =
ψ
(•, •, •) is denoted
G
(
ψ
), and, in Proposition 3.6. we prove
where
G
1
(
ψ
) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra
of type
G
2
is defined in terms of the alternating 3-form
ψ
alone without the need to specify an invariant quadratic form. In the real case the result is more striking. |
doi_str_mv | 10.4153/CJM-1983-045-2 |
format | article |
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G
2
as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let
W
be a complex 7-dimensional vector space. Among the alternating 3-forms on
W
there is a connected dense open subset Ψ(
W
) of “maximal” forms. If
ψ
∈ Ψ(
W
) then the subgroup of AUT
C
(
W
) consisting of the invertible complex-linear transformations
S
such that
ψ
(
S•, S•, S•
) =
ψ
(•, •, •) is denoted
G
(
ψ
), and, in Proposition 3.6. we prove
where
G
1
(
ψ
) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra
of type
G
2
is defined in terms of the alternating 3-form
ψ
alone without the need to specify an invariant quadratic form. In the real case the result is more striking.</description><identifier>ISSN: 0008-414X</identifier><identifier>EISSN: 1496-4279</identifier><identifier>DOI: 10.4153/CJM-1983-045-2</identifier><language>eng</language><ispartof>Canadian journal of mathematics, 1983-10, Vol.35 (5), p.776-806</ispartof><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c842-bf4b38dcff7f34b55af2624324f57a1a183f7064c6c611dd4a2753fe7b42621d3</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Herz, Carl</creatorcontrib><title>Alternating 3-Forms and Exceptional Simple Lie Groups of Type G 2</title><title>Canadian journal of mathematics</title><description>It is now customary to give concrete descriptions of the exceptional simple Lie groups of type
G
2
as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let
W
be a complex 7-dimensional vector space. Among the alternating 3-forms on
W
there is a connected dense open subset Ψ(
W
) of “maximal” forms. If
ψ
∈ Ψ(
W
) then the subgroup of AUT
C
(
W
) consisting of the invertible complex-linear transformations
S
such that
ψ
(
S•, S•, S•
) =
ψ
(•, •, •) is denoted
G
(
ψ
), and, in Proposition 3.6. we prove
where
G
1
(
ψ
) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra
of type
G
2
is defined in terms of the alternating 3-form
ψ
alone without the need to specify an invariant quadratic form. In the real case the result is more striking.</description><issn>0008-414X</issn><issn>1496-4279</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1983</creationdate><recordtype>article</recordtype><recordid>eNotz8tKxDAYBeAgCtbRreu8QMZc_iTtsgwz40jFhV24C2maSKU3kgrO29tBV4cDhwMfQo-MboFJ8bR7eSWsyAWhIAm_QhmDQhHgurhGGaU0J8Dg4xbdpfS1VqEky1BZ9ouPo1268RMLcpjikLAdW7z_cX5eumm0PX7vhrn3uOo8Psbpe054Crg-z2vF_B7dBNsn__CfG1Qf9vXumVRvx9OurIjLgZMmQCPy1oWgg4BGShu44iA4BKktsywXQVMFTjnFWNuC5VqK4HUD6461YoO2f7cuTilFH8wcu8HGs2HUXPxm9ZuL36x-w8Uv691MKA</recordid><startdate>19831001</startdate><enddate>19831001</enddate><creator>Herz, Carl</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19831001</creationdate><title>Alternating 3-Forms and Exceptional Simple Lie Groups of Type G 2</title><author>Herz, Carl</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c842-bf4b38dcff7f34b55af2624324f57a1a183f7064c6c611dd4a2753fe7b42621d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1983</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Herz, Carl</creatorcontrib><collection>CrossRef</collection><jtitle>Canadian journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Herz, Carl</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Alternating 3-Forms and Exceptional Simple Lie Groups of Type G 2</atitle><jtitle>Canadian journal of mathematics</jtitle><date>1983-10-01</date><risdate>1983</risdate><volume>35</volume><issue>5</issue><spage>776</spage><epage>806</epage><pages>776-806</pages><issn>0008-414X</issn><eissn>1496-4279</eissn><abstract>It is now customary to give concrete descriptions of the exceptional simple Lie groups of type
G
2
as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let
W
be a complex 7-dimensional vector space. Among the alternating 3-forms on
W
there is a connected dense open subset Ψ(
W
) of “maximal” forms. If
ψ
∈ Ψ(
W
) then the subgroup of AUT
C
(
W
) consisting of the invertible complex-linear transformations
S
such that
ψ
(
S•, S•, S•
) =
ψ
(•, •, •) is denoted
G
(
ψ
), and, in Proposition 3.6. we prove
where
G
1
(
ψ
) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra
of type
G
2
is defined in terms of the alternating 3-form
ψ
alone without the need to specify an invariant quadratic form. In the real case the result is more striking.</abstract><doi>10.4153/CJM-1983-045-2</doi><tpages>31</tpages></addata></record> |
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ispartof | Canadian journal of mathematics, 1983-10, Vol.35 (5), p.776-806 |
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language | eng |
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source | Freely Accessible Science Journals |
title | Alternating 3-Forms and Exceptional Simple Lie Groups of Type G 2 |
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