# Special unitary group
The **special unitary group** SU(n) is the group of n × n unitary matrices with determinant 1, under matrix multiplication; equivalently, it is the subgroup of the unitary group U(n) defined by \( \mathrm{SU}(n) = \{ U \in \mathrm{U}(n) : \det U = 1 \} \)[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html). It is a connected compact [Lie group](https://www.edgepedia.org/lie-group) of real dimension \( n^{2} - 1 \), also called the unitary unimodular group[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html). The name breaks down as the "U" for unitary (the matrices preserve the standard Hermitian inner product, satisfying \( U^{*}U = I \)) and the "S" for special, the determinant-one condition[2](https://books.physics.oregonstate.edu/GELG/su2.html).
| Key fact | Detail |
|---|---|
| Definition | \( \mathrm{SU}(n) = \{ U \in \mathrm{U}(n) : \det U = 1 \} \), n × n unitary matrices with determinant 1[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) |
| Type | Connected compact Lie group, also called the unitary unimodular group[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) |
| Dimension | Real dimension \( n^{2} - 1 \), one less than U(n)[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) • [3](https://math.ucr.edu/home/baez/lie/node6.html) |
| Lie algebra | \( \mathfrak{su}(n) \), the anti-Hermitian n × n complex matrices with trace zero[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) |
| Center | Scalar matrices \( \zeta I \) with \( \zeta \) an nth root of unity; the quotient is PSU(n)[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) |
| Topology | Simply connected (fundamental group trivial)[4](https://www.maths.dur.ac.uk/users/jamie.j.mason/RepresentationTheoryNotes-epiphany/Ch1.S7.html) |
| Physics role | Gauge group of the Standard Model in the product \( \mathrm{SU}(3) \times \mathrm{SU}(2)_{L} \times \mathrm{U}(1)_{Y} \)[5](https://e-publishing.cern.ch/index.php/CYRSP/article/download/1801/1448/12423) |
## Definition and relation to U(n)
A unitary matrix is a complex matrix satisfying \( U^{*}U = I \), where \( U^{*} \) is the conjugate transpose. Such matrices preserve lengths and inner products, and their determinants satisfy \( |\det U| = 1 \), so the determinant lies on the unit circle in the complex plane[3](https://math.ucr.edu/home/baez/lie/node6.html) • [6](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/MathMeth/mmlecturenotes.pdf). Requiring the determinant to equal exactly 1 selects a subgroup, since the determinant of a product is the product of determinants[7](https://www.chaos.org.uk/~eddy/math/linear/unitary.html).
The determinant-one condition removes exactly one continuous parameter from U(n), which has \( n^{2} \) real parameters; SU(n) therefore has \( n^{2} - 1 \)[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) • [8](https://sites.ualberta.ca/~vbouchar/MAPH464/section-Lie-groups.html). This differs from the orthogonal case, where O(n) and SO(n) have the same dimension, because an orthogonal matrix can only have determinant ±1, a discrete condition[3](https://math.ucr.edu/home/baez/lie/node6.html).
The determinant map expresses how the two groups fit together. Every unitary matrix A can be written as \( A = \zeta U \), where \( \zeta \) is a complex phase solving \( \zeta^{n} = \det A \) and U is special unitary[6](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/MathMeth/mmlecturenotes.pdf). Formally, the determinant gives the exact sequence
\[ 1 \to \mathrm{SU}(n) \to \mathrm{U}(n) \xrightarrow{\det} \mathrm{U}(1) \to 1, \]
meaning SU(n) is the kernel of the determinant homomorphism from U(n) onto the circle group U(1)[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html). Equivalently, \( \mathrm{SU}(n) = \mathrm{U}(n) \cap \mathrm{SL}(n, \mathbb{C}) \), the intersection of the unitary group with the special linear group of determinant-one matrices[9](https://www.sciencedirect.com/topics/mathematics/special-linear-group).
## Lie algebra
The [Lie algebra](https://www.edgepedia.org/lie-algebra) \( \mathfrak{su}(n) \) consists of the anti-Hermitian n × n complex matrices with trace zero:
\[ \mathfrak{su}(n) = \{ X \in M_{n}(\mathbb{C}) : X^{*} = -X, \ \mathrm{tr}\, X = 0 \}. \]
Anti-Hermitian matrices are exactly those for which \( iX \) is Hermitian, and the trace-zero condition implements the determinant-one constraint, since the derivative of the determinant at the identity is the trace[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html) • [3](https://math.ucr.edu/home/baez/lie/node6.html). These traceless anti-Hermitian matrices depend on \( n^{2} - 1 \) real parameters, matching the dimension of the group, and a basis of this space gives the generators of the group[10](https://amslaurea.unibo.it/id/eprint/7171/1/Benassi_Costanza_tesi.pdf).
In the classification of compact simple Lie algebras, \( \mathfrak{su}(n) \) is the member of the infinite family \( A_{n} \), one of the four classical families alongside \( \mathfrak{so} \) and \( \mathfrak{sp} \) types[11](https://math.ucr.edu/home/baez/week90.html). Correspondingly, the [Dynkin diagram](https://www.edgepedia.org/dynkin-diagram) \( A_{n} \) labels the compact Lie group SU(n+1)[12](https://math.ucr.edu/home/baez/week230.html).
## Topology and the SU(2) example
SU(n) is simply connected: every loop in the group can be continuously shrunk to a point, so its [fundamental group](https://www.edgepedia.org/fundamental-group) is trivial[4](https://www.maths.dur.ac.uk/users/jamie.j.mason/RepresentationTheoryNotes-epiphany/Ch1.S7.html). This matters because a simply connected Lie group is determined by its Lie algebra, and homomorphisms from it are in one-to-one correspondence with homomorphisms of Lie algebras[4](https://www.maths.dur.ac.uk/users/jamie.j.mason/RepresentationTheoryNotes-epiphany/Ch1.S7.html).
**SU(2) is the best-understood case.** Writing out the defining conditions for a 2 × 2 matrix shows that
\[ \mathrm{SU}(2) = \left\{ \begin{pmatrix} \alpha & \beta \\ -\beta^{*} & \alpha^{*} \end{pmatrix} : \alpha, \beta \in \mathbb{C}, \ |\alpha|^{2} + |\beta|^{2} = 1 \right\}, \]
which identifies the group with the three-sphere \( S^{3} \) in four real dimensions[6](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/MathMeth/mmlecturenotes.pdf). The same matrices can be written \( a\,\mathbf{1} + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k} \) with \( a^{2} + b^{2} + c^{2} + d^{2} = 1 \), where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) satisfy the quaternion multiplication table, so SU(2) is isomorphic to the group of quaternions of norm 1[3](https://math.ucr.edu/home/baez/lie/node6.html).
SU(2) is thus a double cover of SO(3): two elements of SU(2) correspond to each rotation. The two groups are not isomorphic, but their Lie algebras \( \mathfrak{su}(2) \) and \( \mathfrak{so}(3) \) are, via the identification of traceless anti-Hermitian 2 × 2 matrices with \( \mathbb{R}^{3} \)[3](https://math.ucr.edu/home/baez/lie/node6.html).
## Center and projective quotient
The center of SU(n), the set of elements commuting with everything, consists of the scalar matrices \( \zeta I \) where \( \zeta \) is an nth root of unity, so the center has n elements[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html). Factoring the group by its center gives the projective special unitary group PSU(n)[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html). This quotient is natural in quantum mechanics, where two unitary matrices differing by a global phase represent the same physical operation, which is why one may restrict attention to determinant-one matrices[13](https://qubit.guide/3.4-unitaries-as-rotations).
## Relation to SL(n, C) and representations
SU(n) is the compact real form of the complex Lie group SL(n, C): complexifying SU(n) returns SL(n, C), the group of complex determinant-one matrices[14](https://math.ucr.edu/home/baez/week178.html) • [15](https://math.ucr.edu/home/baez/twf_young.pdf). The two groups exchange advantages. SU(2) is compact, so its finite-dimensional continuous representations can always be made unitary; SL(2, C) is a complex manifold, allowing analytic representations. For SU(2) and SL(2) the finite-dimensional continuous unitary representations of the former correspond precisely to the finite-dimensional analytic representations of the latter, a correspondence [Hermann Weyl](https://www.edgepedia.org/hermann-weyl) generalized: every simply-connected compact simple Lie group has a complex Lie group related to it in this way[16](https://math.ucr.edu/home/baez/history4.pdf).
The real-algebraic irreducible representations of SU(N) are obtained by restricting algebraic irreducible representations of SL(N, C), and they correspond one-to-one with Young diagrams having fewer than N rows[15](https://math.ucr.edu/home/baez/twf_young.pdf).
## Finite special unitary groups
The notation \( \mathrm{SU}_{n}(q) \) denotes a finite special unitary group, whose elements are matrices over the finite field \( \mathbb{F}_{q^{2}} \) that preserve a nondegenerate Hermitian form and have determinant 1. Its order is
\[ |\mathrm{SU}_{n}(q)| = q^{n(n-1)/2} \prod_{j=2}^{n} \left( q^{j} - (-1)^{j} \right). \]
[1](https://mathworld.wolfram.com/SpecialUnitaryGroup.html)
## Role in physics
The gauge groups of fundamental physics are compact Lie groups, and the [Standard Model](https://www.edgepedia.org/standard-model) is built on the product \( \mathrm{SU}(3) \times \mathrm{SU}(2)_{L} \times \mathrm{U}(1)_{Y} \)[5](https://e-publishing.cern.ch/index.php/CYRSP/article/download/1801/1448/12423). [Quantum chromodynamics](https://www.edgepedia.org/quantum-chromodynamics), the theory of the strong interaction, is an SU(3) local gauge theory in which the charges are called color; experimental evidence indicates the number of colors \( N_{c} = 3 \)[5](https://e-publishing.cern.ch/index.php/CYRSP/article/download/1801/1448/12423). The non-[Abelian group](https://www.edgepedia.org/abelian-group) structure of SU(3) implies that gluons interact with themselves, unlike photons[5](https://e-publishing.cern.ch/index.php/CYRSP/article/download/1801/1448/12423).
In quantum computing, single-qubit gates form U(2), but global phases are unobservable, so gates are effectively taken in SU(2)[13](https://qubit.guide/3.4-unitaries-as-rotations) • [17](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513136/). Universality results are phrased in these terms: a set of gates is universal if it generates transformations dense in \( \mathrm{SU}(2^{n}) \), and the Solovay-Kitaev theorem states that a finite gate set dense in SU(2) approximates any SU(2) transformation within \( \varepsilon \) using \( O(\log^{c}(1/\varepsilon)) \) gates[18](https://math.mit.edu/~shor/18.435/oldlectures/lecture8.html).
## References
1. [Special Unitary Group, Wolfram MathWorld](https://mathworld.wolfram.com/SpecialUnitaryGroup.html)
2. [SU(2), Group Theory in Physics, Oregon State University](https://books.physics.oregonstate.edu/GELG/su2.html)
3. [Unitary Matrices, John Baez, Lie Groups course notes](https://math.ucr.edu/home/baez/lie/node6.html)
4. [Representation Theory, Lecture 7: Linear Lie groups and their Lie algebras, Durham University](https://www.maths.dur.ac.uk/users/jamie.j.mason/RepresentationTheoryNotes-epiphany/Ch1.S7.html)
5. [CERN Yellow Reports: School Proceedings, CERN-2025-009](https://e-publishing.cern.ch/index.php/CYRSP/article/download/1801/1448/12423)
6. [Mathematical Methods lecture notes, André Lukas, University of Oxford](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/MathMeth/mmlecturenotes.pdf)
7. [Unitary Transformations, Eddy](https://www.chaos.org.uk/~eddy/math/linear/unitary.html)
8. [Lie groups, MAPH 464 notes, University of Alberta](https://sites.ualberta.ca/~vbouchar/MAPH464/section-Lie-groups.html)
9. [Special Linear Group, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/special-linear-group)
10. [SU(n) thesis, Costanza Benassi, University of Bologna](https://amslaurea.unibo.it/id/eprint/7171/1/Benassi_Costanza_tesi.pdf)
11. [This Week's Finds 90, John Baez](https://math.ucr.edu/home/baez/week90.html)
12. [This Week's Finds 230, John Baez](https://math.ucr.edu/home/baez/week230.html)
13. [Unitaries as rotations, Qubit Guide](https://qubit.guide/3.4-unitaries-as-rotations)
14. [This Week's Finds 178, John Baez](https://math.ucr.edu/home/baez/week178.html)
15. [This Week's Finds (Young), John Baez](https://math.ucr.edu/home/baez/twf_young.pdf)
16. [A Prehistory of n-Categorical Physics, John Baez](https://math.ucr.edu/home/baez/history4.pdf)
17. [SU(2) Decomposition for Quantum Information Dynamics, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC7513136/)
18. [Quantum Computation, Lecture 7, Peter Shor, MIT 18.435](https://math.mit.edu/~shor/18.435/oldlectures/lecture8.html)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Lie algebra structure*
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