Unitary group

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

relative to a form $f$

The group $\def\U{ {\rm U}}\U_n(K,f)$ of all linear transformations $\def\phi{\varphi}\phi$ of an $n$-dimensional right linear space $V$ over a skew-field $K$, preserving a fixed non-singular sesquilinear (relative to an involution $J$ on $K$) form $f$ on $V$, i.e. a $\phi$ such that

$$f(\phi(v),\phi(u)) =f(v,u),\quad v,u\in V.$$ A unitary group is a classical group. Particular cases of unitary groups are a symplectic group (in this case $K$ is a field, $J=1$ and $f$ is an alternating bilinear form) and an orthogonal group ($K$ is a field, ${\rm char}\; K \ne 2$, $J=1$ and $f$ is a symmetric bilinear form). Henceforth, suppose that $J\ne 1$ and that $f$ possesses property $(T)$ (cf. Witt theorem). Multiplying $f$ by a suitable scalar, one can, without changing the unitary group, arrange that $f$ is a Hermitian form, and moreover, by changing $J$, that $f$ is skew-Hermitian.

If one excludes the case $n=2$, $K=\F_4$, then every element of $\U_n(K,f)$ can be written as a product of at most $n+1$ pseudo-reflections (i.e. transformations fixing all elements of some non-isotropic hyperplane in $V$). The centre $Z_n$ of $\U_n(K,f)$ consists of all homotheties of $V$ of the form $\def\g{\gamma}x\mapsto x\g$, $\g\in K$, $\g^J\g =1$.

Let $\nu$ be the Witt index of the form $f$. If $\nu \ne 0$, it will be convenient to take $f$ skew-Hermitian. Let $\def\T{ {\rm T}}\T_n(K,f)$ be the normal subgroup of $\U_n(K,f)$ generated by the unitary transvections, i.e. by the linear transformations of the form $x\mapsto x+a\def\l{\lambda}\l f(a,x)$, where $a$ is an isotropic vector in $V$ and $\l\in S = \{\g\in K : \g^J = \g\}$. The centre of the group $\T_n(K,f)$ is $W_n = \T_n(K,f)\cap Z_n$. The quotient group $\T_n(K,f)/W_n$ is simple for $n\ge 2$, provided $K\ne \F_4,\F_9$. The structure of the quotient group $\U_n(K,f)/\T_n(K,f)$ may be described as follows. Let $\def\S{\Sigma}\S$ be the subgroup of the multiplicative group $K^*$ of $K$ generated by $K^*\cap S$ and let $\def\Om{\Omega}\Om$ be the subgroup of $K^*$ generated by the elements $\l\in K^*$ with the following property: In $V$ there exists a hyperbolic plane (i.e. a non-isotropic two-dimensional subspace containing an isotropic vector) such that $f(v,v)=\l-\l^J$ for a certain vector $v\in V$ orthogonal to the given plane. This subgroup is normal in $K^*$. Let $[k^*,\Om]$ be the subgroup of $K^*$ generated by the commutators $\l w\l^{-1}w^{-1}$, $\l\in K^*$, $w\in \Om$. If one excludes the case $n=3$, $K=\F_4$, then $\U_n(K,f)/\T_n(K,f)$ is isomorphic to $K^*/\S[K^*,\Om]$ for $n\ge 2$.

In many cases the group $\T_n(K,f)$ coincides with the commutator subgroup of $\U_n(K,f)$; this is true, for example, if $\nu\ge 2$. If $K$ is commutative and $\nu\ge 2$, then $\T_n(K,f)$ coincides with the normal subgroup $\U_n^+(K,f)$ of all elements with Dieudonné determinant (cf. Determinant) equal to 1 (excluding the case $n=3, K=\F_4$, $\U_n(K,f)$). The relation between $\U_n(K,f)$ and $\T_n(K,f)$ has also been studied in the case when the skew-field $K$ is finite dimensional over its centre [Di].

Suppose now that $\nu=0$. Then many of the stated results no longer hold (there are examples of unitary groups having an infinite series of normal subgroups with Abelian factors, examples of unitary groups for which $n=2$ and $\U_n^+(K,f)$ does not coincide with the commutator subgroup, etc.). The case most studied is that of locally compact skew-fields of characteristic $\ne 2$ and algebraic number fields.

One of the basic results on automorphisms of unitary groups is the following (cf. [Di]): If ${\rm char}\; K \ne 2$ and $n\ge 3$, then every automorphism of the unitary group $\U_n(K,f)$ has the form $\phi(u) = \chi(u)gug^{-1}$, $u\in \U_n(K,f)$, where $\chi$ is a homomorphism of $\U_n(K,f)$ into its centre $Z_n$ and $g$ a unitary semi-similitude of $V$ (i.e. a bijective semi-linear mapping $V\to V$ satisfying the condition $\def\s{\sigma}f(g(x),g(y)) = r_g(f(x,y))^\s$, where $x,y\in K^*$, $r_g\in K^*$ and $\s$ is the automorphism of $K$ associated with $g$). If $n$ is even, $n\ge 6$, $K$ is a field of characteristic $\ne 2$ and $\nu\ge 1$, then every automorphism of $\U_n^+(K,f)$ is induced by an automorphism of $\U_n(K,f)$.

If $K=\C$, $J$ is the automorphism of complex conjugation and the Hermitian form $f$ is positive definite, then the unitary group $\U_n(K,f)$ is denoted by $\U_n$; it is a real compact connected Lie group and is often simply called the unitary group. In the case of an indefinite form $f$ the group $\U_n(\C,f)$ is often called pseudo-unitary. By the choice of a basis in $V$, $\U_n$ may be identified with the group of all unitary matrices (cf. Unitary matrix). In that case the group $\U_n^+(K,f)$ is called the special unitary group and is denoted by ${\rm SU}_n$.

References

[Bo]	N. Bourbaki, "Algèbre", Eléments de mathématiques, Hermann (1952–1959) pp. Chapts. 7–9 MR2325344 {{MR|2325344} Zbl 1245.16001 Zbl 05948094 Zbl 1107.13001 Zbl 1139.12001
[Di]	J.A. Dieudonné, "La géométrie des groups classiques", Springer (1963) Zbl 0221.20056
[Di2]	J. Dieudonné, "On the automorphisms of the classical groups" Mem. Amer. Math. Soc., 2 (1951) MR0045125 Zbl 0042.25603
[We]	H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
[Za]	A.E. Zalesskii, "Linear groups" Russian Math. Surveys, 36 : 5 (1981) pp. 63–128 Uspekhi Mat. Nauk, 36 : 5 (1981) pp. 57–107 MR0637434