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A function in two variables on a [[Module|module]] (for example, on a vector space) which is linear in one variable and semi-linear in the other. More precisely, a sesquilinear form on a unitary module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847101.png" /> over an associative-commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847102.png" /> with an identity, equipped with an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847103.png" />, is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847105.png" />, linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847106.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847107.png" />, and semi-linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847108.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s0847109.png" /> (see [[Semi-linear mapping|Semi-linear mapping]]). Analogously one defines a sesquilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471013.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471014.png" />-modules. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471016.png" />), one obtains the notion of a [[Bilinear form|bilinear form]] (or a [[Bilinear mapping|bilinear mapping]]). Another important example of a sesquilinear form is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471017.png" /> is a vector space over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471019.png" />. Special cases of sesquilinear forms are Hermitian forms (cf. [[Hermitian form|Hermitian form]]) (and also skew-Hermitian forms).
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{{MSC|15}}
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{{TEX|done}}
  
Sesquilinear forms can also be considered on modules over a non-commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471020.png" />; in this case it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471021.png" /> is an anti-automorphism, that is,
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A ''sesquilinear form'' is a function in two variables on a
 +
[[Module|module]] (for example, on a vector space) which is linear in one variable and semi-linear in the other. More precisely, a sesquilinear form on a unitary module $E$ over an associative-commutative ring $A$ with an identity, equipped with an automorphism $\def\s{\sigma}a\mapsto a^\s$, is a mapping $q:E\times E\to A$, $(x,y)\mapsto q(x,y)$, linear in $x$ for fixed $y$, and semi-linear in $y$ for fixed $x$ (see
 +
[[Semi-linear mapping|Semi-linear mapping]]). Analogously one defines a sesquilinear mapping $E\times F\to G$, where $E$, $F$, $G$ are $A$-modules. In the case when $a^\s = a$ ($a\in A$), one obtains the notion of a
 +
[[Bilinear form|bilinear form]] (or a
 +
[[Bilinear mapping|bilinear mapping]]). Another important example of a sesquilinear form is obtained when $V$ is a vector space over the field $\C$ and $a^\s=\bar a$ is [[complex conjugation]]. Special cases of sesquilinear forms are Hermitian forms (cf. [[Hermitian form]]) (and also skew-Hermitian forms).
  
<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/s/s084/s084710/s08471022.png" /></td> </tr></table>
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Sesquilinear forms can also be considered on modules over a non-commutative ring $A$; in this case it is assumed that $\s$ is an [[anti-automorphism]], that is,
  
 +
$$(ab)^\s = b^\s a^\s\quad a,b\in A.$$
 
For sesquilinear forms it is possible to introduce many notions of the theory of bilinear forms, for example the notions of an orthogonal submodule, a left and a right kernel, a non-degenerate form, the matrix of the form in a given basis, the rank of the form, and conjugate homomorphisms.
 
For sesquilinear forms it is possible to introduce many notions of the theory of bilinear forms, for example the notions of an orthogonal submodule, a left and a right kernel, a non-degenerate form, the matrix of the form in a given basis, the rank of the form, and conjugate homomorphisms.
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Hermann  (1942–1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR></table>
 
  
  
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471023.png" /> be a division ring with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471025.png" /> a right vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471026.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471027.png" /> be an anti-automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471028.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471029.png" /> is an automorphism of the underlying additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471031.png" />. A sesquilinear form relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471033.png" /> is a bi-additive mapping
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Let $D$ be a division ring with centre $k$ and $V$ a right vector space over $D$. Let $\s$ be an anti-automorphism of $D$, i.e. $\s$ is an automorphism of the underlying additive group of $D$ and $\s(xy) =\s(y)\s(x)$. A sesquilinear form relative to $\s$ on $V$ is a bi-additive mapping
 
 
<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/s/s084/s084710/s08471034.png" /></td> </tr></table>
 
  
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$$f : V\times V\to D$$
 
such that
 
such that
  
<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/s/s084/s084710/s08471035.png" /></td> </tr></table>
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$$f(vx,wy) = \s(x)f(v,w)y.$$
 
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Unless $f=0$, the anti-automorphism $\s$ is obviously uniquely determined by $f$.
Unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471036.png" />, the anti-automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471037.png" /> is obviously uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471038.png" />.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471039.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471041.png" />-Hermitian form is a sesquilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471042.png" /> such that moreover
 
  
<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/s/s084/s084710/s08471043.png" /></td> </tr></table>
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Let $\def\e{\epsilon}\e\in k\setminus \{0\}$. A $(\s,\e)$-Hermitian form is a sesquilinear form on $V$ such that moreover
  
One must then have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471045.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471046.png" />. The concepts of a Hermitian, anti-Hermitian, symmetric, anti-symmetric, or bilinear form (or matrix) for complex vector spaces (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471047.png" /> complex conjugation) arise as the special cases of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471048.png" />-Hermitian form, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471049.png" />-Hermitian form, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471050.png" />-Hermitian form, and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471051.png" /> Hermitian form.
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$$f(w,v)=\s (f(v,w))\e$$
 +
One must then have $\e\s(\e)=1$ and $\s^2(x)=\e x\e^{-1}$ for all $x\in D$. The concepts of a Hermitian, anti-Hermitian, symmetric, anti-symmetric, or bilinear form (or matrix) for complex vector spaces (with $\s = $ complex conjugation) arise as the special cases of a $(\s,1)$-Hermitian form, a $(\s,-1)$-Hermitian form, an $({\rm id},1)$-Hermitian form, and an $({\rm id},-1)$ Hermitian form.
  
Given a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471053.png" />. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471054.png" /> is totally isotropic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084710/s08471055.png" />. The Witt index of a sesquilinear form is the dimension of a maximal totally-isotropic subspace.
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Given a subspace $W\subset V$, $W^\perp = \{v\in V : f(v,w)=0 \textrm{ for all } w\in W\}$. A subspace $W$ is totally isotropic if $W\subset W^\perp$. The Witt index of a sesquilinear form is the dimension of a maximal totally-isotropic subspace.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Tits,  "Buildings and BN-pairs of spherical type" , Springer (1974pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.A. Dieudonné,  "La géométrie des groups classiques" , Springer  (1963)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki,  "Algèbre", ''Eléments de mathématiques'', '''2''', Hermann (1942–1959{{MR|0011070}}  {{ZBL|0060.06808}}
 +
|-
 +
|valign="top"|{{Ref|Di}}||valign="top"| J.A. Dieudonné,  "La géométrie des groupes classiques", Springer  (1963)   {{ZBL|0221.20056}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"|  S. Lang,  "Algebra", Addison-Wesley  (1984)  {{MR|0799862}} {{MR|0783636}} {{MR|0760079}}  {{ZBL|0712.00001}}
 +
|-
 +
|valign="top"|{{Ref|Ti}}||valign="top"|  J. Tits,  "Buildings and BN-pairs of spherical type", Springer  (1974)  pp. Chapt. 8  {{ZBL|0295.20047}}
 +
|-
 +
|}

Latest revision as of 13:14, 7 April 2023

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

A sesquilinear form is a function in two variables on a module (for example, on a vector space) which is linear in one variable and semi-linear in the other. More precisely, a sesquilinear form on a unitary module $E$ over an associative-commutative ring $A$ with an identity, equipped with an automorphism $\def\s{\sigma}a\mapsto a^\s$, is a mapping $q:E\times E\to A$, $(x,y)\mapsto q(x,y)$, linear in $x$ for fixed $y$, and semi-linear in $y$ for fixed $x$ (see Semi-linear mapping). Analogously one defines a sesquilinear mapping $E\times F\to G$, where $E$, $F$, $G$ are $A$-modules. In the case when $a^\s = a$ ($a\in A$), one obtains the notion of a bilinear form (or a bilinear mapping). Another important example of a sesquilinear form is obtained when $V$ is a vector space over the field $\C$ and $a^\s=\bar a$ is complex conjugation. Special cases of sesquilinear forms are Hermitian forms (cf. Hermitian form) (and also skew-Hermitian forms).

Sesquilinear forms can also be considered on modules over a non-commutative ring $A$; in this case it is assumed that $\s$ is an anti-automorphism, that is,

$$(ab)^\s = b^\s a^\s\quad a,b\in A.$$ For sesquilinear forms it is possible to introduce many notions of the theory of bilinear forms, for example the notions of an orthogonal submodule, a left and a right kernel, a non-degenerate form, the matrix of the form in a given basis, the rank of the form, and conjugate homomorphisms.



Comments

Let $D$ be a division ring with centre $k$ and $V$ a right vector space over $D$. Let $\s$ be an anti-automorphism of $D$, i.e. $\s$ is an automorphism of the underlying additive group of $D$ and $\s(xy) =\s(y)\s(x)$. A sesquilinear form relative to $\s$ on $V$ is a bi-additive mapping

$$f : V\times V\to D$$ such that

$$f(vx,wy) = \s(x)f(v,w)y.$$ Unless $f=0$, the anti-automorphism $\s$ is obviously uniquely determined by $f$.

Let $\def\e{\epsilon}\e\in k\setminus \{0\}$. A $(\s,\e)$-Hermitian form is a sesquilinear form on $V$ such that moreover

$$f(w,v)=\s (f(v,w))\e$$ One must then have $\e\s(\e)=1$ and $\s^2(x)=\e x\e^{-1}$ for all $x\in D$. The concepts of a Hermitian, anti-Hermitian, symmetric, anti-symmetric, or bilinear form (or matrix) for complex vector spaces (with $\s = $ complex conjugation) arise as the special cases of a $(\s,1)$-Hermitian form, a $(\s,-1)$-Hermitian form, an $({\rm id},1)$-Hermitian form, and an $({\rm id},-1)$ Hermitian form.

Given a subspace $W\subset V$, $W^\perp = \{v\in V : f(v,w)=0 \textrm{ for all } w\in W\}$. A subspace $W$ is totally isotropic if $W\subset W^\perp$. The Witt index of a sesquilinear form is the dimension of a maximal totally-isotropic subspace.

References

[Bo] N. Bourbaki, "Algèbre", Eléments de mathématiques, 2, Hermann (1942–1959) MR0011070 Zbl 0060.06808
[Di] J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1963) Zbl 0221.20056
[La] S. Lang, "Algebra", Addison-Wesley (1984) MR0799862 MR0783636 MR0760079 Zbl 0712.00001
[Ti] J. Tits, "Buildings and BN-pairs of spherical type", Springer (1974) pp. Chapt. 8 Zbl 0295.20047
How to Cite This Entry:
Sesquilinear form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sesquilinear_form&oldid=13338
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article