Unitary group ($U(n)$)

The unitary group $U(n)$ consists of complex matrices preserving the standard Hermitian inner product.

The last-column map has fibre the subgroup fixing the last standard unit vector. Unitary completion gives surjectivity, and compactness gives the quotient homeomorphism. Projecting the other standard vectors onto the orthogonal complement of a nearby unit vector and applying Gram-Schmidt gives a local section, so the projection is a fiber bundle. The bundle $U(n)\to U(n+1)\to S^{2n+1}$ then gives ${\pi }_{i}U(n)\cong {\pi }_{i}U(n+1)$ for $i\le 2n-1$ because both adjacent sphere homotopy groups vanish.

The smooth map $F(A)=A^{\ast }A$ from complex matrices, regarded as a real vector space, to Hermitian matrices has derivative $D{F}_A(K)=A^{\ast }K+K^{\ast }A$. At a unitary $A$, taking $K=AH/2$ gives any prescribed Hermitian $H$. Thus $I$ is a regular value and $F^{-1}(I)=U(n)$ is a smooth manifold of real dimension $n^2$.

// Target: algebra.bigb

For an orthogonal projection $P$ on a complex Hilbert space, this is the group of unitary operators whose off-diagonal blocks relative to $PH\oplus (I-P)H$ are Hilbert-Schmidt operators. Its usual topology combines strong operator convergence with Hilbert-Schmidt convergence of $[P,u]$. It is the group implementable in the corresponding polarized fermionic Fock space.

The quotient identifies unitary operators differing by a scalar phase. A continuous projective unitary representation takes values in this quotient, with the quotient of the strong operator topology. A local continuous section fixes the phase by requiring a chosen nonzero matrix coefficient to be positive real.

The pullback of $U(H)\to PU(H)$ along a continuous circle homomorphism is a compact central extension of the circle. Its lifts commute: a scalar commutator is a continuous alternating bicharacter, which vanishes on a dense cyclic subgroup and hence everywhere. Average a rank-one positive operator over this compact extension. The resulting nonzero compact operator has a finite-dimensional invariant eigenspace, and commuting unitaries have a common eigenvector there. Its continuous character has central weight one; dividing the lifts by that character yields an honest continuous circle representation.

The subgroup $\{U\in U(2n):UJ{U}^T=J\}$, with $J=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$, is the compact symplectic group, often denoted $Sp(n)$ or $USp(2n)$. It preserves both a positive Hermitian metric and a nondegenerate complex alternating form.

The matrix logarithm near the identity maps $Sp(n)=\{A\in U(2n):AJ{A}^t=J\}$ onto an open set of its real compact symplectic Lie algebra $\{X:X^{\ast }=-X,XJ+J{X}^t=0\}$. Its inverse is the matrix exponential. These statements follow by applying the logarithm to $A^{\ast }=A^{-1}$ and $A^t=J^{-1}{A}^{-1}J$, and differentiating $e^{tX}J{e}^{t{X}^t}$. Left translations provide manifold charts at every group element. In block form $X=\left(\begin{array}{cc}P& Q\\ -\bar{Q}& \bar{P}\end{array}\right)$ with $P^{\ast }=-P$ and $Q^t=Q$, giving real dimension $n^2+n(n+1)=n(2n+1)$.

The special unitary group is $SU(n)=\{U\in U(n):\det U=1\}$.

The special unitary group on ${\mathbb{C}}^6$ has dimension $35$. Its complexified Lie algebra is ${\mathfrak{sl}}_6(\mathbb{C})$, with the A5 root system. Its defining representation and exterior square have dimensions $6$ and $15$.

The special unitary group on ${\mathbb{C}}^5$ has dimension $24$. Its complexified Lie algebra is ${\mathfrak{sl}}_5(\mathbb{C})$, with the A4 root system. Its defining representation, exterior square and symmetric square have dimensions $5$, $10$ and $15$.

For the fundamental representation of SU(N), take traceless Hermitian generators with $\mathrm{Tr}(T^a{T}^b)={\delta }^{ab}/2$. Along with the identity matrix, they form an orthogonal basis of all complex matrices. Expanding a matrix in this basis and subtracting its identity component proves the displayed completeness relation. It separates exchanged-index and unchanged-index contributions to color amplitudes.

Near the identity, the matrix logarithm of a unitary matrix is skew-Hermitian. Restrict the logarithm's operator norm below $\pi /n$; determinant one then forces its trace to be zero, since $\det U=\exp (\mathrm{tr}\log U)$ and no nonzero multiple of $2\pi i$ fits this bound. The matrix exponential gives the inverse chart on a ball in the displayed real Lie algebra, of dimension $n^2-1$. Left translation supplies compatible charts on the whole special unitary group.

The four-by-four unitary matrices of determinant one form this special unitary group. Its two-index antisymmetric representation has complex dimension six and the real structure of the SU4 exterior square realizes it on a six-dimensional real space.

For $n\ge 2$, the special unitary group acts transitively on the complex unit sphere. Extend any unit vector to an orthonormal basis, then adjust the phase of a remaining column to give determinant one. The stabilizer subgroup of the first vector consists of special unitary transformations of its orthogonal complement. Thus the sphere is a homogeneous quotient, not just a set with a norm-preserving action.

The SU(3) group consists of unitary complex three-by-three matrices of determinant one. Its defining action on ${\mathbb{C}}^3$ preserves the Hermitian norm. It contains block-diagonal copies of SU(2) and the real SO(3) group. Its Lie algebra consists of traceless anti-Hermitian matrices.

The central element $\omega I$ of SU(3), with $\omega =e^{2\pi i/3}$, acts on an irreducible representation of Dynkin labels $(a,b)$ as ${\omega }^{a+2b}$. This residue is its triality. In a tensor realization the representation has $a+2b$ fundamental boxes, and a complete column of three is a singlet; removing such columns preserves the residue. Tensor products add trialities and complex conjugation reverses them.

A mixed tensor has $k$ defining indices and $l$ dual indices. The SU(3) action applies $U$ to every upper index and $U^{-1}$ to every lower index. Young symmetrizers select permutation types, while invariant Kronecker deltas take traces and invariant alternating tensors convert an antisymmetric pair of defining indices into a dual index, or conversely. Repeated fundamental and dual tensor rules decompose it with multiplicities: $(p,q)\otimes (1,0)=(p+1,q)\oplus (p-1,q+1)\oplus (p,q-1)$ and the conjugate rule, omitting negative labels.

For $T^{abc}$, split its first pair into symmetric and antisymmetric parts. The fully symmetric part $S^{abc}$ is a symmetric cubic representation of SU(3). Write $M^{abc}=T^{(ab)c}-S^{abc}$ and $O^a{}_d={ϵ}_{dbc}{M}^{abc}$. Then $O$ is traceless and $M^{abc}=({ϵ}^{bcd}{O}^a{}_d+{ϵ}^{acd}{O}^b{}_d)/3$. The antisymmetric part is $T^{[ab]c}={ϵ}^{abd}{N}^c{}_d$, with $N^c{}_d={ϵ}_{dab}{T}^{abc}/2$. Splitting $N$ into its traceless part and trace gives the second adjoint representation of SU(3) and the singlet. These maps use invariant tensors, so they commute with the group action and explicitly realize the four irreducible summands.

Symmetrize the $p$ upper and $q$ lower indices separately, and remove every upper-lower trace. The resulting representation has highest weight $p{\omega }_1+q{\omega }_2$. Its dimension isThe subtraction is valid because contraction is onto: in the polynomial realization its adjoint, for the monomial-factorial inner product, is multiplication by the nonzero polynomial ${\sum }_i{x}_i{y}_i$, an injective map. The resulting kernel contains the highest-weight constituent and has precisely its Weyl dimension formula dimension, hence is irreducible. This highest-weight constituent is not always the largest-dimensional constituent of the full mixed tensor.

The degree-three monomials $z_1^a{z}_2^b{z}_3^c$, with $a+b+c=3$, form a basis of this ten-dimensional highest-weight representation. Its highest monomial is $z_1^3$, and $E_{21}=z_2{\partial }_{z_1}$ and $E_{32}=z_3{\partial }_{z_2}$ generate every basis monomial. Its Dynkin labels are $(3,0)$, and the weight of a monomial is $(a-b,b-c)$. In coordinates $(h_1,(h_1+2h_2)/\sqrt{3})$, the ten weights form four horizontal rows of lengths $4,3,2,1$, all with weight multiplicity one. The baryon decuplet realizes this representation of approximate flavor symmetry.

// Target: special-relativity.bigb

For $d(p,q)=(p+1)(q+1)(p+q+2)/2$, the exact largest constituent dimension of a mixed SU(3) tensor representation isEvery constituent highest weight lies below $k{\omega }_1+l{\omega }_2$, so its labels have the form $(p,q)=(k-2a+b,l+a-2b)$ for nonnegative integers $a,b$. If both are positive, reducing each by their minimum increases both $p,q$ and strictly increases the dimension. The resulting boundary labels occur: antisymmetrize disjoint upper pairs into lower indices, or lower pairs into upper indices, then take the symmetric traceless component. This proves the maximum formula. For $(k,l)=(5,0)$, the fully symmetric dimension is 21 but $(3,1)$ has dimension 24. For $k=l$, the fully symmetric traceless dimension $(k+1{)}^3$ is the maximum.

The SU(2) group is the group of two-by-two unitary matrices of determinant one. It is a compact three-dimensional Lie group whose underlying manifold is a three-sphere.

For $\epsilon =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, every two-by-two matrix satisfies $U^T\epsilon U=(\det U)\epsilon$. Thus ${\varphi }^T\epsilon \chi$ is exactly invariant under a common SU(2) group transformation. This is a bilinear contraction without complex conjugation. It vanishes for equal commuting spinors and is the alternating tensor defining the spin singlet state. Overall bosonic exchange symmetry must still be enforced if the spinors label identical particles.

An SU(2) matrix is a two-by-two complex unitary matrix with determinant one. It has the form $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$. Its action on the Riemann sphere is a rotational Möbius transformation; opposite matrices induce the same sphere rotation.

Every element of $SU(2)$ has the unique formThus $SU(2)$ is diffeomorphic to the unit three-sphere in ${\mathbb{C}}^2$ and is a parallelizable manifold.

The determinant $U(n)\to S^1$ sends a unitary loop to a circle-valued loop. A nonzero determinant winding number proves that the original loop class has infinite order.