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''of a non-generate quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867601.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867602.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867603.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867604.png" />) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867605.png" />''
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{{MSC|20}}
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{{TEX|done}}
  
A connected [[Linear algebraic group|linear algebraic group]] which is the simply-connected covering of the irreducible component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867606.png" /> of the identity of the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867607.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867608.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s0867609.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676010.png" /> coincides with the special orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676011.png" />. The spinor group is constructed in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676012.png" /> be the [[Clifford algebra|Clifford algebra]] of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676013.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676015.png" />) be the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676016.png" /> generated by products of an even (odd) number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676017.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676018.png" /> be the canonical anti-automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676019.png" /> defined by the formula
+
The ''spinor group'' or ''spin group'' is associated to
 +
a non-degenerate [[Quadratic form|quadratic form]] $Q$ on an $n$-dimensional vector space $V$ ($n\ge 3$) over a field $k$.
  
<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/s086/s086760/s08676020.png" /></td> </tr></table>
+
It is a connected
 +
[[Linear algebraic group|linear algebraic group]] which is the simply-connected covering of the irreducible component $\def\O{ {\rm O}}\O_n^+(Q)$ of the identity of the orthogonal group $\def\O{ {\rm O}}\O_n(Q)$ of the form $Q$. If $\def\char{ {\rm char}\;}\char k \ne 2$, then $\O_n^+(Q)$ coincides with the special orthogonal group $\def\SO{ {\rm SO}}\SO_n(Q)$. The spinor group is constructed in the following way. Let $C=C(Q)$ be the
 +
[[Clifford algebra|Clifford algebra]] of the pair $(V,Q)$, let $C^+$ ($C^-$) be the subspace of $C$ generated by products of an even (odd) number of elements of $V$, and let $\def\b{\beta}\b$ be the canonical [[anti-automorphism]] of $C$ defined by the formula
  
The inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676021.png" /> enables one to define the Clifford group
+
$$\b(v_1v_2\dots v_n) = v_n\dots v_2v_1.$$
 
+
The inclusion $V\subset C$ enables one to define the Clifford group
<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/s086/s086760/s08676022.png" /></td> </tr></table>
 
  
 +
$$G=\{s\in C : s \textrm{ is invertible in } C \textrm{ and } sVs^{-1} = V\}$$
 
and the even (or special) Clifford group
 
and the even (or special) Clifford group
  
<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/s086/s086760/s08676023.png" /></td> </tr></table>
+
$$G^+ = G\cap C^+.$$
 
+
The spinor group $\def\Spin{ {\rm Spin}}\Spin = \Spin_n(Q) $ is defined by
The spinor group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676024.png" /> is defined by
 
 
 
<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/s086/s086760/s08676025.png" /></td> </tr></table>
 
 
 
The spinor group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676026.png" /> is a quasi-simple (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676027.png" />), connected, simply-connected, linear algebraic group, of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676028.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676029.png" /> and of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676030.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676031.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676032.png" /> it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676033.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676034.png" /> it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676035.png" />. The following isomorphisms hold:
 
 
 
<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/s086/s086760/s08676036.png" /></td> </tr></table>
 
 
 
<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/s086/s086760/s08676037.png" /></td> </tr></table>
 
 
 
There is a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676040.png" /> defined by
 
 
 
<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/s086/s086760/s08676041.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676042.png" />,
 
 
 
<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/s086/s086760/s08676043.png" /></td> </tr></table>
 
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676044.png" /> has a faithful linear representation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676045.png" /> (see [[Spinor representation|Spinor representation]]).
+
$$\Spin_n = \{s\in G^+ : s\b s^{-1} = 1 \}.$$
 +
The spinor group $\Spin_n$ is a quasi-simple (when $n\ne 4$), connected, simply-connected, linear algebraic group, of type $B_m$ when $n=2m+1$ and of type $D_m$ when $n=2m \ge 8$; if $n=6$ it is $A_3$ and if $n=4$ it is $A_1\times A_1$. The following isomorphisms hold:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676046.png" /> is the field of real numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676047.png" /> is positive (or negative) definite, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676048.png" /> of real points of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676049.png" /> is sometimes also called a spinor group. This is a connected simply-connected compact Lie group which is a two-sheeted covering of the special orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676050.png" />. The following isomorphisms hold:
+
$$\Spin_3\simeq \def\SL{ {\rm SL}}\SL_2,\qquad \Spin_2 \simeq \SL_2\times \SL_2,$$
  
<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/s086/s086760/s08676051.png" /></td> </tr></table>
+
$$\Spin_5 \simeq \def\Sp{ {\rm Sp}}\Sp_4,\qquad \Spin_6 \simeq \SL_4.$$
 +
There is a linear representation $\def\th{\vartheta}\th$ of $\Spin_n$ in $V$ defined by
  
<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/s086/s086760/s08676052.png" /></td> </tr></table>
+
$$\th(s)v = svs^{-1},\quad s\in\Spin_n,\; v\in V.$$
 +
If $\char k \ne 2$,
  
(see [[Symplectic group|Symplectic group]]),
+
$$\th(\Spin_n(Q)) = \O_n^+(Q) \textrm{ and } {\rm Ker}\;\th = \{\pm1\}.$$
 +
The group $\Spin_n$ has a faithful linear representation in $C^+$ (see
 +
[[Spinor representation|Spinor representation]]).
  
<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/s086/s086760/s08676053.png" /></td> </tr></table>
+
If $k=\R$ is the field of real numbers and $Q$ is positive (or negative) definite, then the group $\Spin_n(\R)$ of real points of the algebraic group $\Spin_n$ is sometimes also called a spinor group. This is a connected simply-connected compact Lie group which is a two-sheeted covering of the special orthogonal group $\SO_n(\R)$. The following isomorphisms hold:
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) {{MR|}} {{ZBL|0221.20056}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Cartan, "Leçons sur la théorie des spineurs" , '''2''' , Hermann (1938) {{MR|}} {{ZBL|0022.17101}} {{ZBL|0019.36301}} {{ZBL|64.1382.04}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.M. Postnikov, "Lie groups and Lie algebras" , Moscow (1982) (In Russian) {{MR|0905471}} {{ZBL|0597.22001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946) {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </TD></TR></table>
 
  
 +
$$\Spin_3(\R) \simeq \def\SU{ {\rm SU}}\SU_2,\qquad \Spin_4(\R) \simeq \SU_2\times \SU_2,$$
  
 +
$$\Spin_5(\R) \simeq \Sp(0,2),\qquad \Spin_6(\R) \simeq \SU_4,$$
  
====Comments====
+
where $\Sp(0,2)$ is the compact real form of $\Sp_4(\C)$ as described in
See also [[Quadratic form|Quadratic form]].
+
[[Symplectic group|Symplectic group]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676054.png" /> is the so-called even Clifford algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086760/s08676055.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre. Formes sesquilineares et formes quadratiques" , ''Eléments de mathématiques'' , Hermann (1959) pp. Chapt. 9 {{MR|0174550}} {{MR|0107661}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954) {{MR|0060497}} {{ZBL|0057.25901}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Algèbre. Formes sesquilineares et formes quadratiques", ''Eléments de mathématiques'', Hermann (1959) pp. Chapt. 9 {{MR|0174550}} {{MR|0107661}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|BrToDi}}||valign="top"| Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups", Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}}
 +
|-
 +
|valign="top"|{{Ref|Ca}}||valign="top"| E. Cartan, "Leçons sur la théorie des spineurs", '''2''', Hermann (1938) {{MR|}} {{ZBL|0022.17101}} {{ZBL|0019.36301}} {{ZBL|64.1382.04}}
 +
|-
 +
|valign="top"|{{Ref|Ch}}||valign="top"| C. Chevalley, "Theory of Lie groups", '''1''', Princeton Univ. Press (1946) {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}}
 +
|-
 +
|valign="top"|{{Ref|Ch2}}||valign="top"| C. Chevalley, "The algebraic theory of spinors", Columbia Univ. Press (1954) {{MR|0060497}} {{ZBL|0057.25901}}
 +
|-
 +
|valign="top"|{{Ref|Di}}||valign="top"| J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) {{MR|}} {{ZBL|0221.20056}}
 +
|-
 +
|valign="top"|{{Ref|Po}}||valign="top"| M.M. Postnikov, "Lie groups and Lie algebras", Moscow (1982) (In Russian) {{MR|0905471}} {{ZBL|0597.22001}}
 +
|-
 +
|valign="top"|{{Ref|We}}||valign="top"| H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}}
 +
|-
 +
|}

Latest revision as of 13:16, 7 April 2023

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

The spinor group or spin group is associated to a non-degenerate quadratic form $Q$ on an $n$-dimensional vector space $V$ ($n\ge 3$) over a field $k$.

It is a connected linear algebraic group which is the simply-connected covering of the irreducible component $\def\O{ {\rm O}}\O_n^+(Q)$ of the identity of the orthogonal group $\def\O{ {\rm O}}\O_n(Q)$ of the form $Q$. If $\def\char{ {\rm char}\;}\char k \ne 2$, then $\O_n^+(Q)$ coincides with the special orthogonal group $\def\SO{ {\rm SO}}\SO_n(Q)$. The spinor group is constructed in the following way. Let $C=C(Q)$ be the Clifford algebra of the pair $(V,Q)$, let $C^+$ ($C^-$) be the subspace of $C$ generated by products of an even (odd) number of elements of $V$, and let $\def\b{\beta}\b$ be the canonical anti-automorphism of $C$ defined by the formula

$$\b(v_1v_2\dots v_n) = v_n\dots v_2v_1.$$ The inclusion $V\subset C$ enables one to define the Clifford group

$$G=\{s\in C : s \textrm{ is invertible in } C \textrm{ and } sVs^{-1} = V\}$$ and the even (or special) Clifford group

$$G^+ = G\cap C^+.$$ The spinor group $\def\Spin{ {\rm Spin}}\Spin = \Spin_n(Q) $ is defined by

$$\Spin_n = \{s\in G^+ : s\b s^{-1} = 1 \}.$$ The spinor group $\Spin_n$ is a quasi-simple (when $n\ne 4$), connected, simply-connected, linear algebraic group, of type $B_m$ when $n=2m+1$ and of type $D_m$ when $n=2m \ge 8$; if $n=6$ it is $A_3$ and if $n=4$ it is $A_1\times A_1$. The following isomorphisms hold:

$$\Spin_3\simeq \def\SL{ {\rm SL}}\SL_2,\qquad \Spin_2 \simeq \SL_2\times \SL_2,$$

$$\Spin_5 \simeq \def\Sp{ {\rm Sp}}\Sp_4,\qquad \Spin_6 \simeq \SL_4.$$ There is a linear representation $\def\th{\vartheta}\th$ of $\Spin_n$ in $V$ defined by

$$\th(s)v = svs^{-1},\quad s\in\Spin_n,\; v\in V.$$ If $\char k \ne 2$,

$$\th(\Spin_n(Q)) = \O_n^+(Q) \textrm{ and } {\rm Ker}\;\th = \{\pm1\}.$$ The group $\Spin_n$ has a faithful linear representation in $C^+$ (see Spinor representation).

If $k=\R$ is the field of real numbers and $Q$ is positive (or negative) definite, then the group $\Spin_n(\R)$ of real points of the algebraic group $\Spin_n$ is sometimes also called a spinor group. This is a connected simply-connected compact Lie group which is a two-sheeted covering of the special orthogonal group $\SO_n(\R)$. The following isomorphisms hold:

$$\Spin_3(\R) \simeq \def\SU{ {\rm SU}}\SU_2,\qquad \Spin_4(\R) \simeq \SU_2\times \SU_2,$$

$$\Spin_5(\R) \simeq \Sp(0,2),\qquad \Spin_6(\R) \simeq \SU_4,$$

where $\Sp(0,2)$ is the compact real form of $\Sp_4(\C)$ as described in Symplectic group.


References

[Bo] N. Bourbaki, "Algèbre. Formes sesquilineares et formes quadratiques", Eléments de mathématiques, Hermann (1959) pp. Chapt. 9 MR0174550 MR0107661
[BrToDi] Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups", Springer (1985) MR0781344 Zbl 0581.22009
[Ca] E. Cartan, "Leçons sur la théorie des spineurs", 2, Hermann (1938) Zbl 0022.17101 Zbl 0019.36301 Zbl 64.1382.04
[Ch] C. Chevalley, "Theory of Lie groups", 1, Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[Ch2] C. Chevalley, "The algebraic theory of spinors", Columbia Univ. Press (1954) MR0060497 Zbl 0057.25901
[Di] J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) Zbl 0221.20056
[Po] M.M. Postnikov, "Lie groups and Lie algebras", Moscow (1982) (In Russian) MR0905471 Zbl 0597.22001
[We] H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
How to Cite This Entry:
Spinor group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spinor_group&oldid=21941
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article