Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is a special collection of n × n unitary matrices. These matrices have the unique property that when multiplied by their complex conjugate transpose, they give the identity matrix. The unitary group is a type of subgroup of the larger general linear group GL(n, C), and it includes a smaller group called the special unitary group, made up of unitary matrices with determinant 1.

For the simplest case where n = 1, the group U(1) is the same as the circle group. This group is related to all complex numbers that have an absolute value of 1, when these numbers are multiplied together. Every larger unitary group U(n) contains copies of this simple group U(1).

The unitary group U(n) is also a type of real Lie group with a size, or dimension, of n². Its study involves special kinds of matrices called n × n skew-Hermitian matrices, and these are examined using a mathematical operation called the commutator. There is also a broader version called the general unitary group, which includes all matrices A such that A^∗A is a nonzero multiple of the identity matrix.

Unitary groups can also be studied over different types of number systems, not just the complex numbers. In the past, the term hyperorthogonal group was sometimes used to describe the unitary group, especially when it was considered over finite fields.

Properties

The determinant of a unitary matrix is a complex number with norm 1, which helps create a special mapping. This mapping leads to a subgroup called the special unitary group, denoted SU(n), which consists of unitary matrices with determinant 1.

For n > 1, the unitary group U(n) is not abelian. Its center consists of scalar matrices related to U(1), showing that the unitary group is reductive but not semisimple.

Main article: special unitary group

Topology

The unitary group U(n) has a special kind of structure called topology, which helps us understand its shape and properties. It is both compact, meaning it fits into a limited space, and connected, meaning you can move from any point to another without leaving the group.

One important fact is that U(n) is not simply connected. This means that there are loops you can make that cannot be shrunk to a single point. The fundamental group of U(n), which describes these loops, is infinite cyclic, similar to the integers. This property comes from the way U(n) relates to smaller groups like SU(n) and U(1).

Related groups

The unitary group connects to several other important groups in mathematics. It is linked to orthogonal, complex, and symplectic structures, meaning that a unitary structure can show properties of all three when they work together.

The unitary group also has special related groups, such as the special unitary group and the projective unitary group, which have their own unique properties and relationships. These groups help mathematicians understand more about the structure of space and symmetry.

Main articles: Special unitary group and Projective unitary group

G-structure: almost Hermitian

In the language of G-structures, a manifold with a U(n)-structure is an almost Hermitian manifold. This means that the manifold has a special kind of geometric structure related to unitary groups.

Generalizations

The unitary group is a concept from mathematics that can be expanded in several interesting ways. In simple terms, these expansions let us explore the unitary group in different settings or with different rules.

One way to generalize is by using different types of mathematical structures called Hermitian forms. These forms help us define new groups related to the unitary group, called indefinite unitary groups.

Another way is by changing the basic number system we use. Instead of regular complex numbers, we can use structures from finite fields, which are like whole number systems with a limited number of elements. This leads to unitary groups defined over these finite fields.

We can also look at the unitary group as part of a broader family of mathematical objects called algebraic groups. This view helps us understand the unitary group in terms of equations and geometric shapes.

Polynomial invariants

The unitary groups help us understand special patterns in math using two important formulas. The first formula adds up pairs of squared variables, like ( u^2 + v^2 ), and continues this pattern. The second formula looks at differences between products of variables, like ( uv - vu ). These formulas are linked to complex numbers and show how unitary groups behave. They work best when the variables do not commute, meaning the order in which we multiply them matters.

Classifying space

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