Unitary group

Unitary groups may also be defined over fields other than the complex numbers.This subgroup is called the special unitary group, denoted SU(n).We then have a short exact sequence of Lie groups: The above map U(n) to U(1) has a section: we can view U(1) as the subgroup of U(n) that are diagonal with eiθ in the upper left corner and 1 on the rest of the diagonal.The center of U(n) is the set of scalar matrices λI with λ ∈ U(1); this follows from Schur's lemma.Since the center of U(n) is a 1-dimensional abelian normal subgroup of U(n), the unitary group is not semisimple, but it is reductive.The unitary group U(n) is endowed with the relative topology as a subset of M(n, C), the set of all n × n complex matrices, which is itself homeomorphic to a 2n2-dimensional Euclidean space.We can therefore write A path in U(n) from the identity to A is then given by The unitary group is not simply connected; the fundamental group of U(n) is infinite cyclic for all n:[1] To see this, note that the above splitting of U(n) as a semidirect product of SU(n) and U(1) induces a topological product structure on U(n), so that Now the first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, whereas SU(n) is simply connected.The Weyl group of U(n) is the symmetric group Sn, acting on the diagonal torus by permuting the entries: The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups: Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (meaning that one uses the same J in the complex structure and the symplectic form, and that this J is orthogonal; writing all the groups as matrix groups fixes a J (which is orthogonal) and ensures compatibility).At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility).On an almost Kähler manifold, one can write this decomposition as h = g + iω, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and ω is the almost symplectic structure.From the point of view of Lie groups, this can partly be explained as follows: O(2n) is the maximal compact subgroup of GL(2n, R), and U(n) is the maximal compact subgroup of both GL(n, C) and Sp(2n).Thus the intersection O(2n) ∩ GL(n, C) or O(2n) ∩ Sp(2n) is the maximal compact subgroup of both of these, so U(n).These are related as by the commutative diagram at right; notably, both projective groups are equal: PSU(n) = PU(n).From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group 2An, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram An, which corresponds to transpose inverse) and the field automorphism of the extension C/R (namely complex conjugation).The classical unitary group is a real form of this group, corresponding to the standard Hermitian form Ψ, which is positive definite.This can be generalized in a number of ways: Analogous to the indefinite orthogonal groups, one can define an indefinite unitary group, by considering the transforms that preserve a given Hermitian form, not necessarily positive definite (but generally taken to be non-degenerate).The non-degenerate assumption is equivalent to p + q = n. In a standard basis, this is represented as a quadratic form as: and as a symmetric form as: The resulting group is denoted U(p,q).This allows one to define a Hermitian form on an Fq2 vector space V, as an Fq-bilinear map[clarification needed] Further, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to whererepresent the coordinates of w, v ∈ V in some particular Fq2-basis of the n-dimensional space V (Grove 2002, Thm.Thus one can define a (unique) unitary group of dimension n for the extension Fq2/Fq, denoted either as U(n, q) or U(n, q2) depending on the author.The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted SU(n, q) or SU(n, q2).More generally, given a field k and a degree-2 separable k-algebra K (which may be a field extension but need not be), one can define unitary groups with respect to this extension.[5] This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.To any quadratic module (M, h, q) defined by a J-sesquilinear form f on M over a form ring (R, Λ) one can associate the unitary group The special case where Λ = Λmax, with J any non-trivial involution (i.e.,The unitary groups are the automorphisms of two polynomials in real non-commutative variables: These are easily seen to be the real and imaginary parts of the complex form