Namespaces
Variants
Actions

Spinor group

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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=53617
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article