Orthogonal group

2020 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]

An orthogonal group is a group of all linear transformations of an $n$-dimensional vector space $V$ over a field $k$ which preserve a fixed non-singular quadratic form $Q$ on $V$ (i.e. linear transformations $\def\phi{\varphi}\phi$ such that $Q(\phi(v))=Q(v)$ for all $v\in V$). An orthogonal group is a classical group. The elements of an orthogonal group are called orthogonal transformations of $V$ (with respect to $Q$), or also automorphisms of the form $Q$. Furthermore, let ${\rm char\;} k\ne 2$ (for orthogonal groups over fields with characteristic 2 see [Di], [2]) and let $f$ be the non-singular symmetric bilinear form on $V$ related to $Q$ by the formula

$$f(u,v)=\frac{1}{2}(Q(u+v) - Q(u) - Q(v)).$$ The orthogonal group then consists of those linear transformations of $V$ that preserve $f$, and is denoted by $\def\O{ {\rm O} }\O_n(k,f)$, or (when one is talking of a specific field $k$ and a specific form $f$) simply by $\O_n$. If $B$ is the matrix of $f$ with respect to some basis of $V$, then the orthogonal group can be identified with the group of all $(n\times n)$-matrices $A$ with coefficients in $k$ such that $A^TBA = B$ (${}^T$ is transposition).

The description of the algebraic structure of an orthogonal group is a classical problem. The determinant of any element from $\O_n$ is equal to 1 or $-1$. Elements with determinant 1 are called rotations; they form a normal subgroup $\O_n^+(k,f)$ (or simply $\O_n^+$) of index 2 in the orthogonal group, called the rotation group. Elements from $\O_n\setminus \O_n^+$ are called inversions. Every rotation (inversion) is the product of an even (odd) number of reflections from $\O_n$.

Let $Z_n$ be the group of all homotheties $\def\a{\alpha}\phi_\a : v\mapsto \a v$, $\a\in k$, $\a\ne 0$, of the space $V$. Then $\O_n\cap Z_n$ is the centre of $\O_n$; it consists of two elements: $\phi_1$ and $\phi_{-1}$. If $n$ is odd, then $\O_n$ is the direct product of its centre and $\O_n^+$. If $n\ge 3$, the centre of $\O_n^+$ is trivial if $n$ is odd, and coincides with the centre of $\O_n$ if $n$ is even. If $n=2$, the group $\O_n^+$ is commutative and is isomorphic either to the multiplicative group $k^*$ of $k$ (when the Witt index $\nu$ of $f$ is equal to 1), or to the group of elements with norm 1 in $k(\sqrt-\Delta)$, where $\Delta$ is the discriminant of $f$ (when $\nu=0$). The commutator subgroup of $\O_n(k,f)$ is denoted by $\def\Om{\Omega}\Om_n(k,f)$, or simply by $\Om_n$; it is generated by the squares of the elements from $\O_n$. When $n\ge 3$, the commutator subgroup of $\O_n^+$ coincides with $\Om_n$. The centre of $\Om_n$ is $\Om_n\cap Z_n$.

Other classical groups related to orthogonal groups include the canonical images of $\O_n^+$ and $\Om_n$ in the projective group; they are denoted by ${\rm P}\O_n^+(k,f)$ and ${\rm P}\Om_n(k,f)$ (or simply by ${\rm P}\O_n^+$ and ${\rm P}\Om_n$) and are isomorphic to $\O_n^+/(\O_n^+\cap Z_n)$ and $\Om_n/(\Om_n\cap Z_n)$, respectively.

The basic classical facts about the algebraic structure describe the successive factors of the following series of normal subgroups of an orthogonal group:

$$\O_n\supset \O_n^+\supset \Om_n\supset \Om_n\cap Z_n \supset \{e\}.$$ The group $\O_n/\O_n^+$ has order 2. Every element in $\O_n/\Om_n$ has order 2, thus this group is defined completely by its cardinal number, and this number can be either infinite or finite of the form $2^\a$ where $\a$ is an integer. The description of the remaining factors depends essentially on the Witt index $\nu$ of the form $f$.

First, let $\nu\ge 1$. Then $\O_n^+/\Om_n \simeq k^*/{k^*}^2$ when $n>2$. This isomorphism is defined by the spinor norm, which defines an epimorphism from $\O_n^+$ on $k^*/{k^*}^2$ with kernel $\Om_n$. The group $\Om_n\cap Z_n$ is non-trivial (and consists of the transformations $\phi_1$ and $\phi_{-1}$) if and only if $n$ is even and $\Delta\in {k^*}^2$. If $n\ge 5$, then the group ${\rm P}\Om_n = \Om_n/(\Om_n\cap Z_n)$ is simple. The cases where $n=3,4$ are studied separately. Namely, ${\rm P}\Om_3 = \Om_3$ is isomorphic to $\def\PSL{ {\rm PSL}}\PSL_2(k)$ (see Special linear group) and is also simple if $k$ has at least 4 elements (the group $\O_3^+$ is isomorphic to the projective group $\def\PGL{ {\rm PGL}}\PGL_2(k)$). When $\nu=1$, the group ${\rm P}\Om_4 = \Om_4$ is isomorphic to the group $\PSL_2(k(\sqrt{\Delta}))$ and is simple (in this case $\Delta\notin k^2$), while when $\nu=2$, the group ${\rm P}\Om_4$ is isomorphic to $\PSL_2(k)\times \PSL_2(k)$ and is not simple. In the particular case when $k = \R$ and $Q$ is a form of signature $(3,1)$, the group ${\rm P}\Om_4 = \Om_4\simeq \PSL_2(\C)$ is called the Lorentz group.

When $\nu = 0$ (i.e. $Q$ is an anisotropic form), these results are not generally true. For example, if $k=\R$ and $Q$ is a positive-definite form, then $\Om_n = O_n^+$, although $\R^*/{\R^*}^2$ consists of two elements; when $k=\Q$, $n=4$, one can have $\Delta\in k^2$, but $\phi_{-1}\notin \Om_4$. When $\nu=0$, the structures of an orthogonal group and its related groups essentially depend on $k$. For example, if $k=\R$, then ${\rm P}\O_n^+$, $n\ge 3$, $n\ne 4$, $\nu=0$, is simple (and ${\rm P}\O_4^+$ is isomorphic to the direct product $\O_3^+ \times \O_3^+$ of two simple groups); if $k$ is the field of $p$-adic numbers and $\nu=0$, there exists in $\O_3$ (and $\O_4$) an infinite normal series with Abelian quotients. Important special cases are when $k$ is a locally compact field or an algebraic number field. If $k$ is the field of $p$-adic numbers, then $n=0$ is impossible when $\nu\ge 5$. If $k$ is an algebraic number field, then there is no such restriction and one of the basic results is that ${\rm P}\Om_n$, when $\nu=0$ and $n\ge 5$, is simple. In this case, the study of orthogonal groups is closely connected with the theory of equivalence of quadratic forms, where one needs the forms obtained from $Q$ by extension of coefficients to the local fields defined by valuations of $k$ (the Hasse principle).

If $k$ is the finite field $\F_q$ of $q$ elements, then an orthogonal group is finite. The order of $\O_n^+$ for $n$ odd is equal to

$$(q^{n-1}-1)q^{n-2}(q^{n-3}-1)q^{n-4}\cdots (q^2-1)q,$$ while when $n=2m$ it is equal to

$$\def\e{\epsilon}(q^{2m-1}-\e q^{m-1})(q^{2m-2}-1)q^{2m-3}\cdots(q^2-1)q,$$ where $\e=1$ if $(-1)^m\Delta\in \F_q^2$ and $\e=-1$ otherwise. These formulas and general facts about orthogonal groups when $\nu\ge 1$ also allow one to calculate the orders of $\Om_n$ and ${\rm P}\Om_n$, since $\nu\ge 1$ when $n\ge 3$, while the order of $k^*/{k^*}^2$ is equal to 2. The group ${\rm P}\Om_n$, $n\ge 5$, is one of the classical simple finite groups (see also Chevalley group).

One of the basic results on automorphisms of orthogonal groups is the following: If $n\ge 3$, then every automorphism $\phi$ of $\O_n$ has the form $\phi(u)=\chi(u)gug^{-1}$, $u\in \O_n$, where $\chi$ is a fixed homomorphism of $\O_n$ into its centre and $g$ is a fixed bijective semi-linear mapping of $V$ onto itself satisfying $Q(g(v))=r_gQ^\sigma(v)$ for all $v\in V$, where $r_g\in k^*$ while $\sigma$ is an automorphism of $k$. If $\nu\ge 1$ and $n\ge 6$, then every automorphism of $\O_n^+$ is induced by an automorphism of $\O_n$ (see [Di], ).

Like the other classical groups, an orthogonal group has a geometric characterization (under certain hypotheses). Indeed, let $ Q$ be an anisotropic form such that $Q(v)\in k^2$ for all $v\in V$. In this case $k$ is a Pythagorean orderable field. For a fixed order of the field $k$, any sequence $((H_s)_{1\le s\le n}$ constructed from a linearly independent basis $((h_s)_{1\le s\le n}$, where $H_s$ is the set of all linear combinations of the form $\def\l{\lambda}\sum_{j=1}^sl_jh_j$, $\l_s\ge 0$, is called an $n$-dimensional chain of incident half-spaces in $V$. The group $\O_n$ has the property of free mobility, i.e. for any two $n$-dimensional chains of half-spaces there exists a unique transformation from $\O_n$ which transforms the first chain into the second. This property characterizes an orthogonal group: If $L$ is any ordered skew-field and $G$ is a subgroup in ${\rm GL_n(L)}$, $n\ge 3$, having the property of free mobility, then $L$ is a Pythagorean field, while $G=\O_n(L,f)$, where $f$ is an anisotropic symmetric bilinear form such that $f(v,v)\in L_1^2$ for any vector $v$.

Let $\def\bk{ {\bar k}}\bk$ be a fixed algebraic closure of the field $k$. The form $f$ extends naturally to a non-singular symmetric bilinear form ${\bar f}$ on $V\otimes_k \bk$, and the orthogonal group $\O_n(\bk,f)$ is a linear algebraic group defined over $k$ with $\O_n(k,f)$ as group of $k$-points. The linear algebraic groups thus defined (for various $f$) are isomorphic over $\bk$ (but in general not over $k$); the corresponding linear algebraic group over $\bk$ is called the orthogonal algebraic group $\O_n(\bk)$. Its subgroup $\O_n^+(\bk,{\bar f})$ is also a linear algebraic group over $\bk$, and is called a properly orthogonal, or special orthogonal algebraic group (notation: $\def\SO{ {\rm SO}}\SO_n(\bk)$); it is the connected component of the identity of $\O_n(\bk)$. The group $\SO_n(\bk)$ is an almost-simple algebraic group (i.e. does not contain infinite algebraic normal subgroups) of type $B_s$ when $n=2s+1$, $s\ge 1$, and of type $D_s$ when $n=2s$, $s\ge 3$. The universal covering group of $\SO_n$ is a spinor group.

If $ k=\R,\C$ or a $p$-adic field, then $\O_n(k,f)$ has a canonical structure of a real, complex or $p$-adic analytic group. The Lie group $\O_n(\R,f)$ is defined up to isomorphism by the signature of the form $f$; if this signature is $(p,q)$, $p+q=n$, then $\O_n(\R,f)$ is denoted by $\O_(p,q)$ and is called a pseudo-orthogonal group. It can be identified with the Lie group of all real $(n\times n)$-matrices $A$ which satisfy

$$A^TI_{p,q}A = I_{p,q}\qquad\textrm{ where }I_{p,q} = \begin{pmatrix}1_p & 0 \\ 0 & -1_q\end{pmatrix}$$ ($1_s$ denotes the unit $(s\times s)$-matrix). The Lie algebra of this group is the Lie algebra of all real $(n\times n)$-matrices $X$ that satisfy the condition $X^TI_{p,q} = -I_{p.q}X$. In the particular case $q=0$, the group $\O(p,q)$ is denoted by $\O(n)$ and is called a real orthogonal group; its Lie algebra consists of all skew-symmetric real $(n\times n)$-matrices. The Lie group $\O(p,q)$ has four connected components when $q\ne 0$, and two connected components when $q=0$. The connected component of the identity is its commutator subgroup, which, when $q=0$, coincides with the subgroup $\def\SO{ {\rm SO}}\SO(n)$ in $\O(n)$ consisting of all transformations with determinant 1. The group $\O(p,q)$ is compact only when $q=0$. The topological invariants of $\SO(n)$ have been studied. One of the classical results is the calculation of the Betti numbers of the manifold $\SO(n)$: Its Poincaré polynomial has the form

$$\prod_{s=1}^m(1+t^{4s-1})$$ when $n=2m+1$, and the form

$$(1+t^{2m-1})\prod_{s=1}^{m-1}(1+t^{4s-1})$$ when $n=2m$. The fundamental group of the manifold $\SO(n)$ is $\Z_2$. The calculation of the higher homotopy groups $\pi_l(\SO(n))$ is directly related to the classification of locally trivial principal $\SO(n)$-fibrations over spheres. An important part in topological $K$-theory is played by the periodicity theorem, according to which, when $N\gg n$, there are the isomorphisms

$$\pi_{n+8}(\O(N)) \simeq \pi_{n}(\O(N));$$ further,

$$\pi_n(\O(N)) \simeq \Z_2$$ if $n=0,1$;

$$\pi_n(\O(N)) \simeq \Z$$ if $n=3,7$; and

$$\pi_n(\O(N)) = 0$$ if $n=2,4,5,6$. The study of the topology of the group $\O(p,q)$ reduces in essence to the previous case, since the connected component of the identity of $\O(p,q)$ is diffeomorphic to the product $\SO(p)\times \SO(q)$ on a Euclidean space.

Comments

A Pythagorean field is a field in which the sum of two squares is again a square.

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