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

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) Zbl 0221.20056
[2]	N. Bourbaki, "Algèbre" , Eléments de mathématiques , Hermann (1952–1959) pp. Chapts. 7–9 MR2333539 MR2327161 MR2325344 MR2284892 MR0682756 MR0573068 MR0271276 MR0274237 MR0240238 MR0213871 MR0260715 MR0194450 MR0155831 MR0217051 MR0171800 MR0132805 MR0174550 MR0107661 MR0172888 MR0098114 Zbl 05948094 Zbl 1105.18001 Zbl 1107.13002 Zbl 1107.13001 Zbl 1139.12001 Zbl 1111.00001 Zbl 1103.13003 Zbl 1103.13002 Zbl 1103.13001 Zbl 1101.13300 Zbl 0547.13002 Zbl 0547.13001 Zbl 0579.13001 Zbl 0498.12001 Zbl 0455.18010 Zbl 0261.00001 Zbl 0238.13002 Zbl 0211.02401 Zbl 0205.06001 Zbl 0145.04504
[3]	J. Dieudonné, "On the automorphisms of the classical groups" Mem. Amer. Math. Soc. , 2 (1951) MR0045125 Zbl 0042.25603
[4]	H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
[5]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Univ. Paris (1955) Zbl 0068.02102
[6]	A.E. Zalesskii, "Linear groups" Russian Math. Surveys , 36 : 5 (1981) pp. 63–128 Uspekhi Mat. Nauk , 36 : 5 (1981) pp. 57–107 MR0640612 MR0637434 Zbl 1188.20007 Zbl 1025.20037 Zbl 1114.20027 Zbl 0961.20039 Zbl 1109.20306 Zbl 0954.20012 Zbl 0355.20044 Zbl 0364.20052 Zbl 0346.20028 Zbl 0333.20038