FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W4214554351> ?p ?o ?g. }

• W4214554351 abstract "We consider a rank one group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G equals mathematical left-angle upper A comma upper B mathematical right-angle> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G = langle A,B rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting cubically on a module <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this means <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket upper V comma upper A comma upper A comma upper A right-bracket equals 0> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>[V,A,A,A] =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket upper V comma upper G comma upper G comma upper G right-bracket not-equals 0> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>[V,G,G,G] ne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We have to distinguish whether the group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A 0 colon equals upper C Subscript upper A Baseline left-parenthesis left-bracket upper V comma upper A right-bracket right-parenthesis intersection upper C Subscript upper A Baseline left-parenthesis upper V slash upper C Subscript upper V Baseline left-parenthesis upper A right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>:=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>V</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>V</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>A_0 :=C_A([V,A]) cap C_A(V/C_V(A))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is trivial or not. We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A 0> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>A_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is trivial, <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> is a rank one group associated to a quadratic Jordan division algebra. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A 0> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>A_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not trivial (which is always the case if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not abelian), then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A 0> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>A_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defines a subgroup <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G 0> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>G_0</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 G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting quadratically on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We will call <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G 0> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>G_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the <italic>quadratic kernel</italic> of <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>. By a result of Timmesfeld we have <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G 0 approximately-equals normal upper S normal upper L Subscript 2 Baseline left-parenthesis upper J comma upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </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>L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G_0 cong mathrm {SL}_2(J,R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a ring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a special quadratic Jordan division algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J subset-of-or-equal-to upper R> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>J subseteq R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding=application/x-tex>J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case <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> is the special unitary group of a pseudo-quadratic form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Witt index <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the first case <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> is the rank one group for a Freudenthal triple system. These results imply that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper V comma upper G right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(V,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a quadratic pair such that no two distinct root groups commute and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal c normal h normal a normal r upper V not-equals 2 comma 3> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>c</mml:mi> <mml:mi mathvariant=normal>h</mml:mi> <mml:mi mathvariant=normal>a</mml:mi> <mml:mi mathvariant=normal>r</mml:mi> </mml:mrow> <mml:mi>V</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>mathrm {char} Vne 2,3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <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> is a unitary group or an exceptional algebraic group." @default.
• W4214554351 created "2022-03-02" @default.
• W4214554351 creator A5042416156 @default.
• W4214554351 date "2022-03-01" @default.
• W4214554351 modified "2023-10-16" @default.
• W4214554351 title "Cubic Action of a Rank one Group" @default.
• W4214554351 cites W1506692979 @default.
• W4214554351 cites W1582291355 @default.
• W4214554351 cites W1602346696 @default.
• W4214554351 cites W1964827884 @default.
• W4214554351 cites W1965801378 @default.
• W4214554351 cites W1967480562 @default.
• W4214554351 cites W1971843409 @default.
• W4214554351 cites W1974583392 @default.
• W4214554351 cites W1975308595 @default.
• W4214554351 cites W1981855893 @default.
• W4214554351 cites W1994421245 @default.
• W4214554351 cites W1995111790 @default.
• W4214554351 cites W2001521577 @default.
• W4214554351 cites W2007773029 @default.
• W4214554351 cites W2011666822 @default.
• W4214554351 cites W2014736120 @default.
• W4214554351 cites W2018102544 @default.
• W4214554351 cites W2019802807 @default.
• W4214554351 cites W2026568940 @default.
• W4214554351 cites W2035874795 @default.
• W4214554351 cites W2038012249 @default.
• W4214554351 cites W2049655659 @default.
• W4214554351 cites W2051715716 @default.
• W4214554351 cites W2077218899 @default.
• W4214554351 cites W2080417128 @default.
• W4214554351 cites W2092366824 @default.
• W4214554351 cites W2115015270 @default.
• W4214554351 cites W2119938031 @default.
• W4214554351 cites W2137385087 @default.
• W4214554351 cites W2137669795 @default.
• W4214554351 cites W2147139897 @default.
• W4214554351 cites W2148326306 @default.
• W4214554351 cites W2153188363 @default.
• W4214554351 cites W2181824293 @default.
• W4214554351 cites W2317372677 @default.
• W4214554351 cites W2338400435 @default.
• W4214554351 cites W2565805561 @default.
• W4214554351 cites W2903521590 @default.
• W4214554351 cites W2963236893 @default.
• W4214554351 cites W2963787781 @default.
• W4214554351 cites W3103444024 @default.
• W4214554351 cites W3146679729 @default.
• W4214554351 cites W4205987890 @default.
• W4214554351 cites W4230289672 @default.
• W4214554351 cites W4233646047 @default.
• W4214554351 cites W4240880503 @default.
• W4214554351 cites W4242300522 @default.
• W4214554351 cites W4243894389 @default.
• W4214554351 cites W4245181464 @default.
• W4214554351 cites W4252614677 @default.
• W4214554351 doi "https://doi.org/10.1090/memo/1356" @default.
• W4214554351 hasPublicationYear "2022" @default.
• W4214554351 type Work @default.
• W4214554351 citedByCount "1" @default.
• W4214554351 countsByYear W42145543512022 @default.
• W4214554351 crossrefType "journal-article" @default.
• W4214554351 hasAuthorship W4214554351A5042416156 @default.
• W4214554351 hasBestOaLocation W42145543511 @default.
• W4214554351 hasConcept C11413529 @default.
• W4214554351 hasConcept C154945302 @default.
• W4214554351 hasConcept C41008148 @default.
• W4214554351 hasConceptScore W4214554351C11413529 @default.
• W4214554351 hasConceptScore W4214554351C154945302 @default.
• W4214554351 hasConceptScore W4214554351C41008148 @default.
• W4214554351 hasIssue "1356" @default.
• W4214554351 hasLocation W42145543511 @default.
• W4214554351 hasLocation W42145543512 @default.
• W4214554351 hasOpenAccess W4214554351 @default.
• W4214554351 hasPrimaryLocation W42145543511 @default.
• W4214554351 hasRelatedWork W2051487156 @default.
• W4214554351 hasRelatedWork W2073681303 @default.
• W4214554351 hasRelatedWork W2317200988 @default.
• W4214554351 hasRelatedWork W2350741829 @default.
• W4214554351 hasRelatedWork W2358668433 @default.
• W4214554351 hasRelatedWork W2376932109 @default.
• W4214554351 hasRelatedWork W2382290278 @default.
• W4214554351 hasRelatedWork W2390279801 @default.
• W4214554351 hasRelatedWork W2748952813 @default.
• W4214554351 hasRelatedWork W2899084033 @default.
• W4214554351 hasVolume "276" @default.
• W4214554351 isParatext "false" @default.
• W4214554351 isRetracted "false" @default.
• W4214554351 workType "article" @default.