Unitary Group

The unitary group of degree n, denoted U( n ), is the group comprised of all n × n unitary matrices . The operation that defines this group is matrix multiplication . This unitary group stands as a subgroup within the larger general linear group GL( n , C ). A further significant subgroup contained within U( n ) is the special unitary group , which specifically includes those unitary matrices whose determinant is precisely 1.

For the most fundamental case, when n equals 1, the group U(1) is essentially equivalent to the circle group . It’s isomorphic to the collection of all complex numbers that possess an absolute value of 1, with the group operation being standard multiplication. It’s worth noting that all unitary groups, regardless of their degree n, contain copies of this fundamental U(1) group.

From a topological perspective, the unitary group U( n ) is classified as a real Lie group . Its dimension is precisely n squared. The associated Lie algebra for U( n ) is composed of n × n skew-Hermitian matrices . The Lie bracket within this algebra is defined by the commutator operation.

There’s also the concept of the general unitary group, sometimes referred to as the group of unitary similitudes. This broader group encompasses all matrices A for which the product A*A results in a non-zero scalar multiple of the identity matrix . This group is essentially the direct product of the unitary group U( n ) and the group comprising all positive scalar multiples of the identity matrix.

While this discussion primarily focuses on unitary groups over the complex numbers, it’s important to acknowledge that unitary groups can be defined over other fields as well. The term “hyperorthogonal group” is an older, somewhat archaic designation for the unitary group, particularly when considered over finite fields .

Properties

The determinant of any unitary matrix is a complex number whose norm is 1. This property allows the determinant to function as a group homomorphism from U( n ) to U(1), mapping each unitary matrix to its determinant.

$$ \det \colon \operatorname{U}(n) \to \operatorname{U}(1) $$

The kernel of this specific homomorphism is comprised of all unitary matrices whose determinant is exactly 1. This collection forms the special unitary group , commonly denoted as SU( n ). Consequently, we can express the relationship between these groups using a short exact sequence of Lie groups:

$$ 1 \to \operatorname{SU}(n) \to \operatorname{U}(n) \to \operatorname{U}(1) \to 1 $$

This sequence illustrates that SU( n ) is a normal subgroup of U( n ), and the quotient group U( n ) / SU( n ) is isomorphic to U(1). Furthermore, there exists a section for the map from U( n ) to U(1). This can be visualized by considering U(1) as a subgroup within U( n ) consisting of diagonal matrices where the top-left entry is $e^{i\theta}$ and all other diagonal entries are 1. This structure implies that U( n ) can be viewed as a semidirect product of SU( n ) and U(1).

For n > 1, the unitary group U( n ) is not an abelian group; the order of matrix multiplication matters. The center of U( n ), which consists of the elements that commute with all other elements in the group, is precisely the set of scalar matrices $\lambda I$, where $\lambda$ is any complex number from U(1). This follows from the fundamental Schur’s lemma . Therefore, the center is isomorphic to U(1). Since the center of U( n ) is a 1-dimensional abelian normal subgroup, U( n ) itself is not semisimple ; however, it is classified as a reductive group .

Topology

The unitary group U( n ) is equipped with the relative topology . This topology is inherited from its embedding as a subset within M( n , C ), the space of all n × n complex matrices. M( n , C ) itself is topologically equivalent to a Euclidean space of dimension $2n^2$.

As a topological space, U( n ) possesses two key properties: it is both compact and connected . The proof of connectivity often involves demonstrating that any unitary matrix can be transformed into a diagonal form through unitary conjugation. Specifically, any unitary matrix A can be diagonalized by another unitary matrix S, such that $A = S D S^{-1}$, where D is a diagonal matrix. The diagonal entries of D must be complex numbers with an absolute value of 1, which can be expressed as $e^{i\theta_k}$ for $k=1, \dots, n$.

$$ A = S \operatorname{diag} \left(e^{i\theta _{1}}, \dots, e^{i\theta _{n}}\right) S^{-1} $$

A continuous path within U( n ) connecting the identity matrix to A can then be constructed as follows:

$$ t \mapsto S \operatorname{diag} \left(e^{it\theta _{1}}, \dots, e^{it\theta _{n}}\right) S^{-1} $$

This path demonstrates that every unitary matrix is path-connected to the identity, thus establishing the connectivity of U( n ).

However, U( n ) is not simply connected . Its fundamental group, denoted $\pi_1(\operatorname{U}(n))$, is isomorphic to the infinite cyclic group $\mathbb{Z}$ for all $n \ge 1$.

$$ \pi_1(\operatorname{U}(n)) \cong \mathbb{Z} $$

This can be understood by revisiting the semidirect product decomposition of U( n ) as SU( n ) × U(1). This topological product structure induces a product structure on their fundamental groups:

$$ \pi_1(\operatorname{U}(n)) \cong \pi_1(\operatorname{SU}(n)) \times \pi_1(\operatorname{U}(1)) $$

The fundamental group of U(1), which is topologically a circle , is known to be $\mathbb{Z}$. Meanwhile, SU( n ) is simply connected, meaning its fundamental group is trivial ($\pi_1(\operatorname{SU}(n)) = {e}$). Therefore, the fundamental group of U( n ) is isomorphic to $\mathbb{Z}$. The determinant map, det: U( n ) → U(1), plays a crucial role here, inducing an isomorphism between the fundamental groups of U( n ) and U(1). The splitting map U(1) → U( n ) acts as the inverse isomorphism.

The Weyl group associated with U( n ) is the symmetric group $S_n$. This group acts on the diagonal torus by permuting the entries of the diagonal matrices. For a diagonal unitary matrix with entries $e^{i\theta_k}$, the action of a permutation $\sigma \in S_n$ results in a matrix with entries $e^{i\theta_{\sigma(k)}}$:

$$ \operatorname{diag} \left(e^{i\theta _{1}}, \dots, e^{i\theta _{n}}\right) \mapsto \operatorname{diag} \left(e^{i\theta _{\sigma (1)}}, \dots, e^{i\theta _{\sigma (n)}}\right) $$

Related Groups

2-out-of-3 Property

A remarkable property of the unitary group is its position as the intersection of three other fundamental groups: the orthogonal group O(2n), the general linear group GL(n, C), and the symplectic group Sp(2n, R).

$$ \operatorname{U}(n) = \operatorname{O}(2n) \cap \operatorname{GL}(n, \mathbf{C}) \cap \operatorname{Sp}(2n, \mathbf{R}) $$

This implies that a unitary structure can be viewed as a compatible combination of an orthogonal structure, a complex structure, and a symplectic structure. Specifically, it means that a unitary structure embodies an orthogonal structure, a complex structure, and a symplectic structure that are mutually consistent. When these groups are considered as matrix groups, a specific complex structure J is fixed, which is also orthogonal, thereby ensuring compatibility.

The “2-out-of-3” property means that the intersection of any two of these three groups yields the third. For instance, a compatible orthogonal and complex structure implies the existence of a symplectic structure, and so on.

At the level of the defining equations, this can be understood as follows:

Any two of these equations logically imply the third.

When viewed in terms of forms, a Hermitian form can be decomposed into its real and imaginary components. The real part is symmetric, corresponding to the orthogonal structure, while the imaginary part is skew-symmetric, related to the symplectic structure. These components are intrinsically linked by the complex structure, which represents their compatibility. On an almost Kähler manifold , this decomposition is expressed as $h = g + i\omega$, where $h$ is the Hermitian form, $g$ is the Riemannian metric , $i$ is the almost complex structure , and $\omega$ is the almost symplectic structure .

From the perspective of Lie groups , this intersection property can be further elucidated. The orthogonal group O(2n) serves as the maximal compact subgroup of GL(2n, R). Similarly, U( n ) is the maximal compact subgroup for both GL( n , C ) and Sp(2n). Consequently, the intersection of O(2n) with either GL( n , C ) or Sp(2n) will result in the maximal compact subgroup of the respective intersection, which is U( n ). From this viewpoint, the less intuitive aspect is the intersection of GL( n , C ) and Sp(2n) yielding U( n ).

Special Unitary and Projective Unitary Groups

Analogous to how the orthogonal group O( n ) has the special orthogonal group SO( n ) as a subgroup and the projective orthogonal group PO( n ) as a quotient, and PSO( n ) as a subquotient , the unitary group U( n ) is associated with the special unitary group SU( n ), the projective unitary group PU( n ), and the projective special unitary group PSU( n ). These groups are interconnected, as depicted in the commutative diagram to the right. Notably, for the classical unitary groups, the projective unitary group and the projective special unitary group are identical: PSU( n ) = PU( n ).

It is important to distinguish that this relationship holds for the classical unitary group over the complex numbers. When considering unitary groups defined over finite fields, analogous special unitary and projective unitary groups can be formed, but it is generally the case that PSU( n , q²) ≠ PU( n , q²).

G-structure: Almost Hermitian

In the framework of G-structures , a manifold endowed with a U( n )-structure is precisely what is termed an almost Hermitian manifold . This connection highlights the deep interplay between group theory and differential geometry.

Generalizations

From the perspective of Lie theory , the classical unitary group U( n ) can be understood as a specific real form of the Steinberg group ²A_n. This Steinberg group is an algebraic group that arises from the combined action of two automorphisms: the diagram automorphism of the general linear group (which effectively reverses the Dynkin diagram A_n, corresponding to the transpose-inverse operation on matrices) and the field automorphism of the extension C/R, which is complex conjugation . Both of these automorphisms are of order 2, commute with each other, and are automorphisms of the algebraic group. The unitary group U( n ) corresponds to the set of fixed points under the product of these two automorphisms, when considered as an algebraic group. The standard positive-definite Hermitian form is instrumental in defining this specific real form.

This fundamental construction can be extended in several significant ways:

• Generalizing Hermitian Forms: By considering Hermitian forms that are not necessarily positive definite, one can define indefinite unitary groups, denoted as U(p, q). These groups play crucial roles in various areas of physics and mathematics.

• Field Extensions: The field extension C/R can be generalized to any separable algebra of degree 2. This opens up the possibility of defining unitary groups over a wider range of algebraic structures, including, notably, degree 2 extensions of finite fields.

• Other Diagrams and Groups of Lie Type: The process of combining automorphisms can be applied to other Dynkin diagrams, leading to different families of algebraic groups. This includes other Steinberg groups such as ²D_n, ²E_6, ³D_4, and also the Suzuki–Ree groups :

  $$ {}^{2}!B_{2}\left(2^{2n+1}\right), {}^{2}!F_{4}\left(2^{2n+1}\right), {}^{2}!G_{2}\left(3^{2n+1}\right) $$

• Algebraic Group Points: By viewing the generalized unitary group as an algebraic group, one can study its points over various k-algebras, providing a more abstract and flexible approach to their study.

Indefinite Forms

Mirroring the construction of indefinite orthogonal groups , indefinite unitary groups can be defined. This is achieved by considering transformations that preserve a specific Hermitian form, which is not necessarily positive definite, although it is typically assumed to be non-degenerate. These considerations are carried out within a vector space over the complex numbers.

Given a Hermitian form $\Psi$ on a complex vector space $V$, the unitary group $U(\Psi)$ is defined as the set of all linear transformations $M$ that preserve the form, meaning $\Psi(Mv, Mw) = \Psi(v, w)$ for all vectors $v, w \in V$. In the matrix representation, where the form is represented by a matrix $\Phi$, this condition translates to $M^* \Phi M = \Phi$.

Similar to how symmetric forms over the real numbers are classified by their signature , Hermitian forms are also determined by their signature. All non-degenerate Hermitian forms on a complex vector space of dimension $n$ are unitarily equivalent to a diagonal form with $p$ entries of $+1$ and $q$ entries of $-1$ on the diagonal, where $p+q=n$. In a standard basis, this is represented by a quadratic form:

$$ \lVert z\rVert {\Psi }^{2}=\lVert z{1}\rVert ^{2}+\dots +\lVert z_{p}\rVert ^{2}-\lVert z_{p+1}\rVert ^{2}-\dots -\lVert z_{n}\rVert ^{2} $$

And as a symmetric form:

$$ \Psi (w,z)={\bar {w}}{1}z{1}+\cdots +{\bar {w}}{p}z{p}-{\bar {w}}{p+1}z{p+1}-\cdots -{\bar {w}}{n}z{n} $$

The resulting group is denoted as $U(p, q)$.

Finite Fields

Over a finite field $\mathbb{F}q$ with $q = p^r$ elements, there exists a unique quadratic extension field, $\mathbb{F}{q^2}$, of order $q^2$. This field possesses a unique automorphism of order 2, denoted by $\alpha$, which maps an element $x$ to $x^q$. This automorphism is the $r$-th power of the Frobenius automorphism . This structure allows for the definition of a Hermitian form on an $\mathbb{F}{q^2}$ vector space $V$. A Hermitian form is an $\mathbb{F}q$-bilinear map $\Psi: V \times V \to \mathbb{F}{q^2}$ that satisfies the conditions $\Psi(w, v) = \alpha(\Psi(v, w))$ and $\Psi(w, cv) = c\Psi(w, v)$ for $c \in \mathbb{F}{q^2}$. Furthermore, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard form, which is represented by the identity matrix. In coordinates, with respect to a suitable basis, this standard form is given by:

$$ \Psi (w,v)=w^{\alpha }\cdot v=\sum {i=1}^{n}w{i}^{q}v_{i} $$

where $w_i, v_i$ are the coordinates of $w, v \in V$. The unitary group associated with this structure is denoted as $U(n, q)$ or $U(n, q^2)$. The subgroup of matrices with determinant 1 is the special unitary group, $SU(n, q)$ or $SU(n, q^2)$. For consistency, this article will adopt the $U(n, q^2)$ convention. The center of $U(n, q^2)$ has order $q+1$ and consists of scalar matrices $cI_V$ such that $c^{q+1} = 1$. The center of the special unitary group $SU(n, q^2)$ has order $\gcd(n, q+1)$ and comprises those unitary scalars whose order divides $n$. The quotient of the unitary group by its center is the projective unitary group , $PU(n, q^2)$, and the quotient of the special unitary group by its center is the projective special unitary group , $PSU(n, q^2)$. For most cases where $n > 1$ and $(n, q^2)$ does not fall into specific exceptional pairs, $SU(n, q^2)$ is a perfect group , and $PSU(n, q^2)$ is a finite simple group .

Degree-2 Separable Algebras

More generally, given a field $k$ and a separable algebra $K$ of degree 2 over $k$ (which might be a field extension or a more complex structure), one can define unitary groups relative to this extension. Such an algebra $K$ possesses a unique $k$-automorphism, denoted by $a \mapsto \bar{a}$, which is an involution (meaning $\bar{\bar{a}} = a$) and fixes precisely the elements of $k$ (i.e., $a = \bar{a}$ if and only if $a \in k$). This generalizes complex conjugation and the conjugation in degree 2 finite field extensions, enabling the definition of Hermitian forms and unitary groups as described previously.

Algebraic Groups

The defining conditions for a unitary group are expressed by polynomial equations. Crucially, these equations are polynomial over the base field $k$, but not necessarily over the extension algebra $K$. For the standard Hermitian form $\Phi = I$, the defining condition in matrix form is $A^* A = I$, where $A^* = \bar{A}^T$ is the conjugate transpose . If a different form $\Phi$ is used, the condition becomes $A^* \Phi A = \Phi$. Consequently, the unitary group is an algebraic group . The points of this algebraic group over a $k$-algebra $R$ are given by:

$$ \operatorname{U}(n, K/k, \Phi)(R) := \left{A \in \operatorname{GL}(n, K \otimes_k R) : A^* \Phi A = \Phi \right} $$

For the specific case of the field extension $\mathbb{C}/\mathbb{R}$ and the standard positive-definite Hermitian form, these equations define an algebraic group whose real and complex points are:

$$ \begin{aligned} \operatorname{U}(n, \mathbf{C}/\mathbf{R})(\mathbf{R}) &= \operatorname{U}(n) \ \operatorname{U}(n, \mathbf{C}/\mathbf{R})(\mathbf{C}) &= \operatorname{GL}(n, \mathbf{C}) \end{aligned} $$

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

Unitary Group of a Quadratic Module

The unitary group of a quadratic module represents a significant generalization of the linear algebraic groups described above. This concept incorporates a wide array of classical algebraic groups as special instances. The foundational work on this generalization can be attributed to Anthony Bak’s thesis.

To define this generalized unitary group, one must first establish the notion of quadratic modules. Let $R$ be a ring equipped with an anti-automorphism $J$. Additionally, let $\varepsilon \in R^\times$ be an invertible element such that $r^{J^2} = \varepsilon r \varepsilon^{-1}$ for all $r \in R$, and $\varepsilon^J = \varepsilon^{-1}$. Two important subsets of $R$ are defined:

$$ \Lambda_{\text{min}} := \left{r \in R \mid r - r^J \varepsilon = 0\right} $$ $$ \Lambda_{\text{max}} := \left{r \in R \mid r^J \varepsilon = -r\right} $$

A subset $\Lambda \subseteq R$ is designated as a form parameter if it satisfies the conditions $\Lambda_{\min} \subseteq \Lambda \subseteq \Lambda_{\max}$ and $r^J \Lambda r \subseteq \Lambda$ for all $r \in R$. A pair $(R, \Lambda)$, where $R$ is a ring and $\Lambda$ is a form parameter, is termed a form ring.

Now, consider an $R$-module $M$ and a $J$-sesquilinear form $f$ on $M$. A $J$-sesquilinear form satisfies $f(xr, ys) = r^J f(x, y) s$ for any $x, y \in M$ and $r, s \in R$. Associated with $f$, two functions are defined: $h(x,y) := f(x,y) + f(y,x)^J \varepsilon \in R$ and $q(x) := f(x,x) \pmod{\Lambda}$. If $h(x,y)$ and $q(x)$ have specific properties related to $\Lambda$, $f$ is said to define a $\Lambda$-quadratic form $(h, q)$ on $M$. A quadratic module over $(R, \Lambda)$ is then defined as a triple $(M, h, q)$ where $M$ is an $R$-module and $(h, q)$ is a $\Lambda$-quadratic form.

To any quadratic module $(M, h, q)$ defined by a $J$-sesquilinear form $f$ on $M$ over a form ring $(R, \Lambda)$, one can associate the unitary group:

$$ U(M) := {\sigma \in \operatorname{GL}(M) \mid \forall x,y \in M, h(\sigma x, \sigma y) = h(x,y) \text{ and } q(\sigma x) = q(x)} $$

A particularly important special case arises when $\Lambda = \Lambda_{\max}$, and $J$ is a non-trivial involution (i.e., $J \neq \operatorname{id}_R$ and $J^2 = \operatorname{id}_R$) with $\varepsilon = -1$. This specific setup recovers the “classical” unitary group, viewed as an algebraic group.

Polynomial Invariants

The unitary groups act as automorphisms on a pair of polynomials defined over real, non-commutative variables. These polynomials are:

$$ C_1 = (u^2 + v^2) + (w^2 + x^2) + (y^2 + z^2) + \dots $$ $$ C_2 = (uv - vu) + (wx - xw) + (yz - zy) + \dots $$

These invariants can be intuitively understood as the real and imaginary parts of the complex form $Z\overline{Z}$. Individually, $C_1$ is an invariant of the orthogonal group O(2n), and $C_2$ is an invariant of the symplectic group Sp(2n). When considered together, they form the invariants of the unitary group U( n ), which is a subgroup of both O(2n) and Sp(2n). The non-commutative nature of the variables is essential for the second polynomial, $C_2$, to be non-trivial; if the variables commuted, $C_2$ would identically be zero.

Classifying Space

The classifying space associated with the unitary group U( n ) is a topic detailed in the article titled “Classifying space for U( n )”. This concept is fundamental in algebraic topology and K-theory.