Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication. All the unitary groups contain copies of this group.

The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator.

The general unitary group (also called the group of unitary similitudes) consists of all matrices A such that AA is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism

The kernel of this homomorphism is the set of unitary matrices with determinant 1. 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 e in the upper left corner and 1 on the rest of the diagonal. Therefore U(n) is a semidirect product of U(1) with SU(n).

The unitary group U(n) is not abelian for n > 1. The center of U(n) is the set of scalar matrices λI with λ ∈ U(1); this follows from Schur's lemma. The center is then isomorphic to U(1). 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.

As a topological space, U(n) is both compact and connected. To show that U(n) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write

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

The determinant map det: U(n) → U(1) induces an isomorphism of fundamental groups, with the splitting U(1) → U(n) inducing the inverse.

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).

In fact, it is the intersection of any two of these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth.[3][4]

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 + , 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). From this perspective, what is unexpected is the intersection GL(n, C) ∩ Sp(2n) = U(n).

Just as the orthogonal group O(n) has the special orthogonal group SO(n) as subgroup and the projective orthogonal group PO(n) as quotient, and the projective special orthogonal group PSO(n) as subquotient, the unitary group U(n) has associated to it the special unitary group SU(n), the projective unitary group PU(n), and the projective special unitary group PSU(n). These are related as by the commutative diagram at right; notably, both projective groups are equal: PSU(n) = PU(n).

In the language of G-structures, a manifold with a U(n)-structure is an almost Hermitian manifold.

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). Here one is working with a vector space over the complex numbers.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U(Ψ) is the group of transforms that preserve the form: the transform M such that Ψ(Mv, Mw) = Ψ(v, w) for all v, wV. In terms of matrices, representing the form by a matrix denoted Φ, this says that MΦM = Φ.

Just as for symmetric forms over the reals, Hermitian forms are determined by signature, and are all unitarily congruent to a diagonal form with p entries of 1 on the diagonal and q entries of −1. The non-degenerate assumption is equivalent to p + q = n. In a standard basis, this is represented as a quadratic form as:

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.

For the field extension C/R and the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by:

The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined, which incorporates as special cases many different classical algebraic groups. The definition goes back to Anthony Bak's thesis.[6]

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 unitary groups are the automorphisms of two polynomials in real non-commutative variables:

The classifying space for U(n) is described in the article Classifying space for U(n).