Spinor group

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

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