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''relative to a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955301.png" />''
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{{MSC|20}}
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{{TEX|done}}
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955302.png" /> of all linear transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955303.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955304.png" />-dimensional right linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955305.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955306.png" />, preserving a fixed non-singular sesquilinear (relative to an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955307.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955308.png" />) form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u0955309.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553010.png" />, i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553011.png" /> such that
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The ''unitary group'' (relative to a form $f$) is
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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
  
<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/u/u095/u095530/u09553012.png" /></td> </tr></table>
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$$f(\phi(v),\phi(u)) =f(v,u),\quad v,u\in V.$$
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A unitary group is a
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[[Classical group|classical group]]. Particular cases of unitary groups are a
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[[Symplectic group|symplectic group]] (in this case $K$ is a field, $J=1$ and $f$ is an alternating
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[[Bilinear form|bilinear form]]) and an
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[[Orthogonal group|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.
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[[Witt theorem|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.
  
A unitary group is a [[Classical group|classical group]]. Particular cases of unitary groups are a [[Symplectic group|symplectic group]] (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553013.png" /> is a field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553015.png" /> is an alternating [[Bilinear form|bilinear form]]) and an [[Orthogonal group|orthogonal group]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553016.png" /> is a field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553019.png" /> is a symmetric bilinear form). Henceforth, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553020.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553021.png" /> possesses property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553022.png" /> (cf. [[Witt theorem|Witt theorem]]). Multiplying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553023.png" /> by a suitable scalar, one can, without changing the unitary group, arrange that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553024.png" /> is a Hermitian form, and moreover, by changing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553025.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553026.png" /> is skew-Hermitian.
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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 $\def\g{\gamma}x\mapsto x\g$, $\g\in K$, $\g^J\g =1$.
  
If one excludes the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553028.png" />, then every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553029.png" /> can be written as a product of at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553030.png" /> pseudo-reflections (i.e. transformations fixing all elements of some non-isotropic hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553031.png" />). The centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553033.png" /> consists of all homotheties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553034.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553037.png" />.
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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 $n\ge 2$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553038.png" /> be the Witt index of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553040.png" />, it will be convenient to take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553041.png" /> skew-Hermitian. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553042.png" /> be the normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553043.png" /> generated by the unitary transvections, i.e. by the linear transformations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553045.png" /> is an isotropic vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553047.png" />. The centre of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553049.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553050.png" /> is simple for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553051.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553052.png" />. The structure of the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553053.png" /> may be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553054.png" /> be the subgroup of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553056.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553057.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553058.png" /> be the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553059.png" /> generated by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553060.png" /> with the following property: In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553061.png" /> there exists a hyperbolic plane (i.e. a non-isotropic two-dimensional subspace containing an isotropic vector) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553062.png" /> for a certain vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553063.png" /> orthogonal to the given plane. This subgroup is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553064.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553065.png" /> be the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553066.png" /> generated by the commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553069.png" />. If one excludes the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553072.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553073.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553074.png" />.
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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.
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[[Determinant|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
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{{Cite|Di}}.
  
In many cases the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553075.png" /> coincides with the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553076.png" />; this is true, for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553078.png" /> is commutative and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553079.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553080.png" /> coincides with the normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553081.png" /> of all elements with Dieudonné determinant (cf. [[Determinant|Determinant]]) equal to 1 (excluding the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553083.png" />). The relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553085.png" /> has also been studied in the case when the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553086.png" /> is finite dimensional over its centre [[#References|[1]]].
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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.
  
Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553087.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553089.png" /> does not coincide with the commutator subgroup, etc.). The case most studied is that of locally compact skew-fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553090.png" /> and algebraic number fields.
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One of the basic results on automorphisms of unitary groups is the following (cf.
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{{Cite|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
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[[Semi-linear mapping|semi-linear mapping]] $V\to V$ satisfying the
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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 $\U_n(K,f)$.
  
One of the basic results on automorphisms of unitary groups is the following (cf. [[#References|[1]]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553092.png" />, then every automorphism of the unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553093.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553096.png" /> is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553097.png" /> into its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u09553099.png" /> a unitary semi-similitude of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530100.png" /> (i.e. a bijective [[Semi-linear mapping|semi-linear mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530101.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530105.png" /> is the automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530106.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530107.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530108.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530110.png" /> is a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530112.png" />, then every automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530113.png" /> is induced by an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530114.png" />.
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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.
 
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[[Unitary matrix|Unitary matrix]]). In that case the group $\U_n^+(K,f)$ is called the special unitary group and is denoted by ${\rm SU}_n$.
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530116.png" /> is the automorphism of complex conjugation and the Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530117.png" /> is positive definite, then the unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530118.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530119.png" />; it is a real compact connected Lie group and is often simply called the unitary group. In the case of an indefinite form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530120.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530121.png" /> is often called pseudo-unitary. By the choice of a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530123.png" /> may be identified with the group of all unitary matrices (cf. [[Unitary matrix|Unitary matrix]]). In that case the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530124.png" /> is called the special unitary group and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095530/u095530125.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Dieudonné,   "La géométrie des groups classiques" , Springer  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Algèbre" , ''Eléments de mathématiques'' , Hermann  (1952–1959) pp. Chapts. 7–9</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"J. Dieudonné,   "On the automorphisms of the classical groups" ''Mem. Amer. Math. Soc.'' , '''2''' (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"H. Weyl,   "The classical groups, their invariants and representations" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Univ. Paris  (1955)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"A.E. Zalesskii,   "Linear groups" ''Russian Math. Surveys'' , '''36''' : 5 (1981) pp. 63–128 ''Uspekhi Mat. Nauk'' , '''36''' : 5 (1981) pp. 57–107</TD></TR></table>
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{|
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|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Algèbre", ''Eléments de mathématiques'', Hermann (1952–1959) pp. Chapts. 7–9 {{MR|2325344}} {{MR|2325344} {{ZBL|1245.16001}} {{ZBL|05948094}}  {{ZBL|1107.13001}} {{ZBL|1139.12001}}
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|valign="top"|{{Ref|Di}}||valign="top"| J.A. Dieudonné, "La géométrie des groups classiques", Springer (1963) {{MR|}} {{ZBL|0221.20056}}
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|valign="top"|{{Ref|Di2}}||valign="top"| J. Dieudonné, "On the automorphisms of the classical groups" ''Mem. Amer. Math. Soc.'', '''2''' (1951) {{MR|0045125}} {{ZBL|0042.25603}}
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|valign="top"|{{Ref|We}}||valign="top"| H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}}
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|-
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|valign="top"|{{Ref|Za}}||valign="top"| A.E. Zalesskii, "Linear groups" ''Russian Math. Surveys'', '''36''' : 5 (1981) pp. 63–128 ''Uspekhi Mat. Nauk'', '''36''' : 5 (1981) pp. 57–107 {{MR|0637434}}
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Latest revision as of 16:59, 30 November 2014

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

The unitary group (relative to a form $f$) is 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
How to Cite This Entry:
Unitary group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_group&oldid=13710
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article