Unitary group

 ( Lie Groups ) 

In mathematics, the unitary group of degree n, denoted U( n), is the group of Unitary matrix, 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 1.

In the simple case , the group U(1) corresponds to the circle group, isomorphic to the set of all that have absolute value 1, under multiplication. All the unitary groups contain copies of this group.

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

The general unitary group, also called the group of unitary similitudes, consists of all matrices A such that A^∗ A 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 .

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

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

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:

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

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^iθ 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 group for . The center of U( n) is the set of scalar matrices λI with ; 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 group.

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Topology The unitary group U( n) is endowed with the relative topology as a subset of , the set of all complex matrices, which is itself homeomorphic to a 2 n²-dimensional Euclidean space.

As a topological space, U( n) is both compact space and connected space. 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

$A\; =\; S\backslash ,\backslash operatorname\{diag\}\backslash left(e^\{i\backslash theta\_1\},\; \backslash dots,\; e^\{i\backslash theta\_n\}\backslash right)\backslash ,S^\{-1\}.$

A path in U( n) from the identity to A is then given by

$t\; \backslash mapsto\; S\; \backslash ,\; \backslash operatorname\{diag\}\backslash left(e^\{it\backslash theta\_1\},\; \backslash dots,\; e^\{it\backslash theta\_n\}\backslash right)\backslash ,S^\{-1\}\; .$

The unitary group is not simply connected; the fundamental group of U( n) is infinite cyclic for all n: Proposition 13.11

$\backslash pi\_1(\backslash operatorname\{U\}(n))\; \backslash cong\; \backslash mathbf\{Z\}\; .$

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

$\backslash pi\_1(\backslash operatorname\{U\}(n))\; \backslash cong\; \backslash pi\_1(\backslash operatorname\{SU\}(n))\; \backslash times\; \backslash pi\_1(\backslash operatorname\{U\}(1)).$

Now the first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, whereas SU( n) is simply connected. Proposition 13.11

The determinant map induces an isomorphism of fundamental groups, with the splitting inducing the inverse.

The Weyl group of U( n) is the symmetric group S _n, acting on the diagonal torus by permuting the entries:

$\backslash operatorname\{diag\}\backslash left(e^\{i\backslash theta\_1\},\; \backslash dots,\; e^\{i\backslash theta\_n\}\backslash right)\; \backslash mapsto\; \backslash operatorname\{diag\}\backslash left(e^\{i\backslash theta\_\{\backslash sigma(1)\}\},\; \backslash dots,\; e^\{i\backslash theta\_\{\backslash sigma(n)\}\}\backslash right)$

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Related groups

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2-out-of-3 property The unitary group is the 3-fold intersection of the orthogonal group, complex, and symplectic group groups:

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

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.

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

$\backslash begin\{array\}\{r|r\}$

 \text{Symplectic} &
 A^\mathsf{T}JA = J \\
 \hline
 \text{Complex} &
 A^{-1}JA = J \\
 \hline
 \text{Orthogonal} &
 A^\mathsf{T} = A^{-1}
     

\end{array}

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, one can write this decomposition as , 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 , this can partly be explained as follows: O(2 n) is the maximal compact subgroup of , and U( n) is the maximal compact subgroup of both and Sp(2 n). Thus the intersection or is the maximal compact subgroup of both of these, so U( n). From this perspective, what is unexpected is the intersection .

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Special unitary and projective unitary groups 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: .

The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general .

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

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Generalizations From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group ²A_n, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram A _n, which corresponds to transpose inverse) and the field automorphism of the extension C/ R (namely complex conjugation). Both these automorphisms are automorphisms of the algebraic group, have order 2, 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 Ψ, which is positive definite.

This can be generalized in a number of ways:

• generalizing to other Hermitian forms yields indefinite unitary groups ;
• the field extension can be replaced by any degree 2 separable algebra, most notably a degree 2 extension of a finite field;
• generalizing to other diagrams yields other groups of Lie type, namely the other Steinberg groups ²D _n, ²E₆, ³D₄, (in addition to ²A _n) and Suzuki–Ree groups
• : $\{\}^2\backslash !B\_2\backslash left(2^\{2n+1\}\backslash right),\; \{\}^2\backslash !F\_4\backslash left(2^\{2n+1\}\backslash right),\; \{\}^2\backslash !G\_2\backslash left(3^\{2n+1\}\backslash right);$
• considering a generalized unitary group as an algebraic group, one can take its points over various algebras.

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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.

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 for all . In terms of matrices, representing the form by a matrix denoted Φ, this says that .

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 . In a standard basis, this is represented as a quadratic form as:

$\backslash lVert\; z\; \backslash rVert\_\backslash Psi^2\; =\; \backslash lVert\; z\_1\; \backslash rVert^2\; +\; \backslash dots\; +\; \backslash lVert\; z\_p\; \backslash rVert^2\; -\; \backslash lVert\; z\_\{p+1\}\; \backslash rVert^2\; -\; \backslash dots\; -\; \backslash lVert\; z\_n\; \backslash rVert^2$

and as a symmetric form as:

$\backslash Psi(w,\; z)\; =\; \backslash bar\; w\_1\; z\_1\; +\; \backslash cdots\; +\; \backslash bar\; w\_p\; z\_p\; -\; \backslash bar\; w\_\{p+1\}z\_\{p+1\}\; -\; \backslash cdots\; -\; \backslash bar\; w\_n\; z\_n.$

The resulting group is denoted .

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Finite fields Over the finite field with elements, F _q, there is a unique quadratic extension field, F _q², with order 2 automorphism $\backslash alpha\backslash colon\; x\; \backslash mapsto\; x^q$ (the rth power of the Frobenius automorphism). This allows one to define a Hermitian form on an F _q² vector space V, as an F _q-bilinear map $\backslash Psi\backslash colon\; V\; \backslash times\; V\; \backslash to\; K$ such that $\backslash Psi(w,\; v)\; =\; \backslash alpha\; \backslash left(\backslash Psi(v,\; w)\backslash right)$ and $\backslash Psi(w,\; cv)\; =\; c\backslash Psi(w,\; v)$ for . Further, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to

$\backslash Psi(w,\; v)\; =\; w^\backslash alpha\; \backslash cdot\; v\; =\; \backslash sum\_\{i=1\}^n\; w\_i^q\; v\_i$

where $w\_i,v\_i$ represent the coordinates of in some particular F _q²-basis of the n-dimensional space V .

Thus one can define a (unique) unitary group of dimension n for the extension F _q²/ F _q, denoted either as or depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted or . For convenience, this article will use the convention. The center of has order and consists of the scalar matrices that are unitary, that is those matrices cI_V with $c^\{q+1\}\; =\; 1$. The center of the special unitary group has order and consists of those unitary scalars which also have order dividing n. 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 . In most cases ( and }), is a perfect group and is a finite simple group, .

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Degree-2 separable algebras 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.

First, there is a unique k-automorphism of K $a\; \backslash mapsto\; \backslash bar\; a$ which is an involution and fixes exactly k ($a\; =\; \backslash bar\{a\}$ if and only if ).Milne, Algebraic Groups and Arithmetic Groups, p. 103 This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.

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Algebraic groups The equations defining a unitary group are polynomial equations over k (but not over K): for the standard form , the equations are given in matrices as , where $A^*\; =\; \backslash bar\{A\}^\backslash mathsf\{T\}$ is the conjugate transpose. Given a different form, they are . The unitary group is thus an algebraic group, whose points over a k-algebra R are given by:

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

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:

$\backslash begin\{align\}$

 \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{align}

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

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Unitary group of a quadratic module 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 group. The definition goes back to Anthony Bak's thesis.Bak, Anthony (1969), "On modules with quadratic forms", Algebraic K-Theory and its Geometric Applications (editors—Moss R. M. F., Thomas C. B.) Lecture Notes in Mathematics, Vol. 108, pp. 55-66, Springer.

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

Let R be a ring with anti-automorphism J, $\backslash varepsilon\; \backslash in\; R^\backslash times$ such that $r^\{J^2\}\; =\; \backslash varepsilon\; r\; \backslash varepsilon^\{-1\}$ for all r in R and $\backslash varepsilon^J\; =\; \backslash varepsilon^\{-1\}$. Define

$\backslash begin\{align\}$

 \Lambda_\text{min} &:= \left\{r \in R \ : \ r - r^J\varepsilon\right\}, \\
 \Lambda_\text{max} &:= \left\{r \in R \ : \ r^J\varepsilon = -r\right\}.
     

\end{align}

Let be an additive subgroup of R, then Λ is called form parameter if $\backslash Lambda\_\backslash text\{min\}\; \backslash subseteq\; \backslash Lambda\; \backslash subseteq\; \backslash Lambda\_\backslash text\{max\}$ and $r^J\; \backslash Lambda\; r\; \backslash subseteq\; \backslash Lambda$. A pair such that R is a ring and Λ a form parameter is called form ring.

Let M be an R-module and f a J-sesquilinear form on M (i.e., $f(xr,\; ys)\; =\; r^J\; f(x,\; y)s$ for any $x,\; y\; \backslash in\; M$ and $r,\; s\; \backslash in\; R$). Define $h(x,\; y)\; :=\; f(x,\; y)\; +\; f(y,\; x)^J\; \backslash varepsilon\; \backslash in\; R$ and $q(x)\; :=\; f(x,\; x)\; \backslash in\; R/\backslash Lambda$, then f is said to define the Λ-quadratic form on M. A quadratic module over is a triple such that M is an R-module and is a Λ-quadratic form.

To any quadratic module defined by a J-sesquilinear form f on M over a form ring one can associate the unitary group

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

The special case where , with J any non-trivial involution (i.e., $J\; \backslash neq\; id\_R,\; J^2\; =\; id\_R$ and gives back the "classical" unitary group (as an algebraic group).

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

$\backslash begin\{align\}$

 C_1 &= \left(u^2 + v^2\right) + \left(w^2 + x^2\right) + \left(y^2 + z^2\right) + \ldots \\
 C_2 &= \left(uv - vu\right) + \left(wx - xw\right) + \left(yz - zy\right) + \ldots
     

\end{align}

These are easily seen to be the real and imaginary parts of the complex form $Z\; \backslash overline\{Z\}$. The two invariants separately are invariants of O(2 n) and Sp(2 n). Combined they make the invariants of U( n) which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.

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

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See also

• Special unitary group
• Projective unitary group
• Orthogonal group
• Symplectic group

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Notes

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