Unitary group

In mathematics, the unitary group of degree n, written as U(n), is a special set of n × n unitary matrices. These matrices have a special property: when you multiply them by their complex conjugate transpose, you get the identity matrix. The unitary group is a type of subgroup of a larger group called the general linear group GL(n, C). It also includes a smaller group known as the special unitary group, which consists 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 connected to all complex numbers that have an absolute value of 1, when these numbers are multiplied together. Every larger unitary group U(n) includes 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 studied using a math 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 using different types of number systems, not just the complex numbers. In the past, the term hyperorthogonal group was sometimes used for the unitary group, especially when it was studied over finite fields.

Properties

The determinant of a unitary matrix is a special kind of number with a value of 1. This helps create a subgroup called the special unitary group, written as SU(n).

For n > 1, the unitary group U(n) is not abelian. Its center is made of special matrices related to U(1). This shows that the unitary group is reductive but not semisimple.

Main article: special unitary group

Topology

The unitary group U(n) has a special structure called topology. This helps us understand its shape and properties. It is compact, meaning it fits into a limited space. It is also connected, so 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 there are loops that cannot be shrunk to a single point. The fundamental group of U(n) describes these loops. It is infinite cyclic, similar to the integers. This property comes from how 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. This means 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. 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 the manifold has a special geometric structure linked to unitary groups.

Generalizations

The unitary group is a math idea that can be stretched in many ways. Simply put, these stretches let us study the unitary group in new places or with new rules.

One way to stretch it is by using different math tools called Hermitian forms. These tools help us build new groups linked to the unitary group, known as indefinite unitary groups.

We can also change the basic number system we use. Instead of regular complex numbers, we can use structures from finite fields. These are like whole number systems with only a few parts. This gives us unitary groups built over these finite fields.

Lastly, we can see the unitary group as part of a bigger group of math objects called algebraic groups. This view helps us understand the unitary group using equations and 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 ). 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.

Classifying space

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