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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757801.png" /> be a division ring (cf. [[Division algebra|Division algebra]]) with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757802.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757803.png" /> be a right vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757804.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757805.png" /> be an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757807.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p0757809.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578010.png" />. Assume also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578013.png" />. Set
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$#C+1 = 27 : ~/encyclopedia/old_files/data/P075/P.0705780 Pseudo\AAhquadratic form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578014.png" /></td> </tr></table>
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This is an additive subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578016.png" /> be the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578017.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578018.png" /> denote the quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578019.png" />. A pseudo-quadratic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578020.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578021.png" /> such that there exists a trace-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578022.png" />-Hermitian form (cf. [[Sesquilinear form|Sesquilinear form]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578024.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578025.png" /> is uniquely determined by this and is called the sesquilinearization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578026.png" />.
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Let $  D $
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be a division ring (cf. [[Division algebra|Division algebra]]) with centre  $  k $
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and let  $  V $
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be a right vector space over  $  D $.  
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Let  $  \sigma $
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be an automorphism of  $  D $
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and  $  \epsilon \in D $
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such that $  \epsilon \sigma ( \epsilon ) = 1 $,
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$  \sigma  ^ {2} ( x) = \epsilon x \epsilon  ^ {-} 1 $
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for all  $  x \in D $.  
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Assume also that  $  \epsilon \not\equiv - 1 $
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if  $  \sigma = \mathop{\rm id} $
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and $  \mathop{\rm char} ( D) \neq 2 $.  
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Set
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075780/p07578027.png" />-pseudo-quadratic form is a [[Quadratic form|quadratic form]] in the usual sense. The Witt index of a pseudo-quadratic form is that of the associated sesquilinear form.
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$$
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D ( \sigma , \epsilon )  = \{ {x - \sigma ( x) \epsilon } : {x \in D } \}
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.
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$$
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This is an additive subgroup of  $  D $.
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Let  $  \overline{D}\; $
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be the quotient  $  \overline{D}\; = D / D ( \sigma , \epsilon ) $,
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and let  $  x \mapsto \overline{x}\; $
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denote the quotient mapping  $  D \rightarrow \overline{D}\; $.
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A pseudo-quadratic form on  $  V $
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is a function  $  q : V \rightarrow \overline{D}\; $
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such that there exists a trace-valued  $  ( \sigma - \epsilon ) $-
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Hermitian form (cf. [[Sesquilinear form|Sesquilinear form]])  $  f :  V \times V \rightarrow D $
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such that  $  q ( v+ w) = q( v) + q( w) + \overline{ {f( v , w) }}\; $.  
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The form  $  f $
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is uniquely determined by this and is called the sesquilinearization of  $  q $.
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A  $  (  \mathop{\rm id} , 1) $-
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pseudo-quadratic form is a [[Quadratic form|quadratic form]] in the usual sense. The Witt index of a pseudo-quadratic form is that of the associated sesquilinear form.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Tits,  "Buildings and BN-pairs of spherical type" , Springer  (1974)  pp. Sect. 8.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Eléments de mathématique. Algèbre" , Hermann  (1959)  pp. Chapt. 9. Formes sesquilinéaires et formes quadratiques</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groups classiques" , Springer  (1963)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Tits,  "Buildings and BN-pairs of spherical type" , Springer  (1974)  pp. Sect. 8.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Eléments de mathématique. Algèbre" , Hermann  (1959)  pp. Chapt. 9. Formes sesquilinéaires et formes quadratiques</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groupes classiques" , Springer  (1963)</TD></TR>
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</table>

Latest revision as of 13:15, 7 April 2023


Let $ D $ be a division ring (cf. Division algebra) with centre $ k $ and let $ V $ be a right vector space over $ D $. Let $ \sigma $ be an automorphism of $ D $ and $ \epsilon \in D $ such that $ \epsilon \sigma ( \epsilon ) = 1 $, $ \sigma ^ {2} ( x) = \epsilon x \epsilon ^ {-} 1 $ for all $ x \in D $. Assume also that $ \epsilon \not\equiv - 1 $ if $ \sigma = \mathop{\rm id} $ and $ \mathop{\rm char} ( D) \neq 2 $. Set

$$ D ( \sigma , \epsilon ) = \{ {x - \sigma ( x) \epsilon } : {x \in D } \} . $$

This is an additive subgroup of $ D $. Let $ \overline{D}\; $ be the quotient $ \overline{D}\; = D / D ( \sigma , \epsilon ) $, and let $ x \mapsto \overline{x}\; $ denote the quotient mapping $ D \rightarrow \overline{D}\; $. A pseudo-quadratic form on $ V $ is a function $ q : V \rightarrow \overline{D}\; $ such that there exists a trace-valued $ ( \sigma - \epsilon ) $- Hermitian form (cf. Sesquilinear form) $ f : V \times V \rightarrow D $ such that $ q ( v+ w) = q( v) + q( w) + \overline{ {f( v , w) }}\; $. The form $ f $ is uniquely determined by this and is called the sesquilinearization of $ q $.

A $ ( \mathop{\rm id} , 1) $- pseudo-quadratic form is a quadratic form in the usual sense. The Witt index of a pseudo-quadratic form is that of the associated sesquilinear form.

References

[a1] J. Tits, "Buildings and BN-pairs of spherical type" , Springer (1974) pp. Sect. 8.2
[a2] N. Bourbaki, "Eléments de mathématique. Algèbre" , Hermann (1959) pp. Chapt. 9. Formes sesquilinéaires et formes quadratiques
[a3] J.A. Dieudonné, "La géométrie des groupes classiques" , Springer (1963)
How to Cite This Entry:
Pseudo-quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-quadratic_form&oldid=15903