Unitary group
relative to a form
The group of all linear transformations
of an
-dimensional right linear space
over a skew-field
, preserving a fixed non-singular sesquilinear (relative to an involution
on
) form
on
, i.e. a
such that
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A unitary group is a classical group. Particular cases of unitary groups are a symplectic group (in this case is a field,
and
is an alternating bilinear form) and an orthogonal group (
is a field,
,
and
is a symmetric bilinear form). Henceforth, suppose that
and that
possesses property
(cf. Witt theorem). Multiplying
by a suitable scalar, one can, without changing the unitary group, arrange that
is a Hermitian form, and moreover, by changing
, that
is skew-Hermitian.
If one excludes the case ,
, then every element of
can be written as a product of at most
pseudo-reflections (i.e. transformations fixing all elements of some non-isotropic hyperplane in
). The centre
of
consists of all homotheties of
of the form
,
,
.
Let be the Witt index of the form
. If
, it will be convenient to take
skew-Hermitian. Let
be the normal subgroup of
generated by the unitary transvections, i.e. by the linear transformations of the form
, where
is an isotropic vector in
and
. The centre of the group
is
. The quotient group
is simple for
, provided
. The structure of the quotient group
may be described as follows. Let
be the subgroup of the multiplicative group
of
generated by
and let
be the subgroup of
generated by the elements
with the following property: In
there exists a hyperbolic plane (i.e. a non-isotropic two-dimensional subspace containing an isotropic vector) such that
for a certain vector
orthogonal to the given plane. This subgroup is normal in
. Let
be the subgroup of
generated by the commutators
,
,
. If one excludes the case
,
, then
is isomorphic to
for
.
In many cases the group coincides with the commutator subgroup of
; this is true, for example, if
. If
is commutative and
, then
coincides with the normal subgroup
of all elements with Dieudonné determinant (cf. Determinant) equal to 1 (excluding the case
,
). The relation between
and
has also been studied in the case when the skew-field
is finite dimensional over its centre [1].
Suppose now that . 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
and
does not coincide with the commutator subgroup, etc.). The case most studied is that of locally compact skew-fields of characteristic
and algebraic number fields.
One of the basic results on automorphisms of unitary groups is the following (cf. [1]): If and
, then every automorphism of the unitary group
has the form
,
, where
is a homomorphism of
into its centre
and
a unitary semi-similitude of
(i.e. a bijective semi-linear mapping
satisfying the condition
, where
,
and
is the automorphism of
associated with
). If
is even,
,
is a field of characteristic
and
, then every automorphism of
is induced by an automorphism of
.
If ,
is the automorphism of complex conjugation and the Hermitian form
is positive definite, then the unitary group
is denoted by
; it is a real compact connected Lie group and is often simply called the unitary group. In the case of an indefinite form
the group
is often called pseudo-unitary. By the choice of a basis in
,
may be identified with the group of all unitary matrices (cf. Unitary matrix). In that case the group
is called the special unitary group and is denoted by
.
References
[1] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1963) |
[2] | N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1952–1959) pp. Chapts. 7–9 |
[3] | J. Dieudonné, "On the automorphisms of the classical groups" Mem. Amer. Math. Soc. , 2 (1951) |
[4] | H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) |
[5] | , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Univ. Paris (1955) |
[6] | A.E. Zalesskii, "Linear groups" Russian Math. Surveys , 36 : 5 (1981) pp. 63–128 Uspekhi Mat. Nauk , 36 : 5 (1981) pp. 57–107 |
Unitary group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_group&oldid=13710