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Wikipedia / Unitary group

In mathematics, the unitary group, is the group unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group, and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant.

corresponds to the circle group, isomorphic to the set of all complex numbers that have absolute value, under multiplication. All the unitary groups contain copies of this group.

is a real Lie group. The Lie algebra 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 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.

Unitary groups may also be defined over fields other than the complex numbers. The hyperorthogonal group is an archaic name for the unitary group, especially over finite fields.

Properties

Since the determinant, the determinant gives a group homomorphism

The kernel. This subgroup is called the special unitary group. We then have a short exact sequence

is a semidirect product.

is not abelian. The center; this follows from Schur's lemma-dimensional abelian normal subgroup, the unitary group is not semisimple, but it is reductive.

Topology

is endowed with the relative topology-dimensional Euclidean space.

is both compact and connected can be diagonalized on the main diagonal

A path

The unitary group is not simply connected:

as a semidirect product

is topologically a circle, which is well known to have a fundamental group is simply connected.

inducing the inverse.

The Weyl group is the symmetric group

Related groups

2-out-of-3 property

The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (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.

At the level of equations, this can be seen as follows:Any two of these equations implies the third.

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 is the Riemannian metric is the almost complex structure is the almost symplectic structure.

From the point of view of Lie groups is the maximal compact subgroup.

Special unitary and projective unitary groups

has the special orthogonal group as subgroup and the projective orthogonal group as quotient, and the projective special orthogonal group as subquotient has associated to it the special unitary group, the projective unitary group, and the projective special unitary group. These are related as by the commutative diagram.

The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields.

G-structure: almost Hermitian

In the language of G-structures-structure is an almost Hermitian manifold.

Generalizations

From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram, which corresponds to transpose inverse) and the field automorphism (namely complex conjugation, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group. The classical unitary group is a real form of this group, corresponding to the standard Hermitian form, which is positive definite.

This can be generalized in a number of ways:

Indefinite forms

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.

.

Just as for symmetric forms over the reals, Hermitian forms are determined by signature, and are all unitarily congruent

.

Finite fields

Over the finite fieldth power of the Frobenius automorphism. Further, all non-degenerate Hermitian forms on a vector space over a finite field.

is called the special unitary group. The quotient of the unitary group by its center is called the projective unitary group, and the quotient of the special unitary group by its center is the projective special unitary group is a perfect group is a finite simple group.

Degree-2 separable algebras

(which may be a field extension but need not be), one can define unitary groups with respect to this extension.

). finite field extensions, and allows one to define Hermitian forms and unitary groups as above.

Algebraic groups

is the conjugate transpose. The unitary group is thus an algebraic group

In fact, the unitary group is a linear algebraic group.

Unitary group of a quadratic module

just defined, which incorporates as special cases many different classical algebraic groups. The definition goes back to Anthony Bak's thesis.

To define it, one has to define quadratic modules first:

is called form parameter a form parameter is called form ring.

is said to define the -quadratic form. A quadratic module-quadratic form.

one can associate the unitary group

gives back the "classical" unitary group (as an algebraic group).

Polynomial invariants

which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.

Classifying space

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

See also

Notes

References