nLab special unitary Lie algebra

Idea

The Lie algebra $\mathfrak{su}(n)$ of the special unitary group $SU(n)$. Canonically identified with matrices with complex number entries that are skew-hermitean and have vanishing trace.

Examples

• su(2)

Related concepts

• unitary Lie algebra

• special orthogonal Lie algebra

• special linear Lie algebra

References

General

Textbook accounts:

• Walter Pfeifer, The Lie algebras $\mathfrak{su}(N)$, Birkhäuser Verlag 2003 (doi:10.1007/978-3-0348-8097-8)

Relation between $\mathfrak{su}(\infty)$ and $Lie(SDiff(\Sigma))$

On the identification of the special unitary Lie algebra $\mathfrak{su}(n)$, as $n \to \infty$, with the Lie algebra of area-preserving diffeomorphisms of surfaces $\Sigma$ (cf. also at Quantization of the M2-brane to the BFSS matrix model):

For the 2-sphere:

• Jens Hoppe; section B.II of: Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, PhD thesis, MIT (1982) [hdl:1721.1/15717, pdf]

• Bernard de Wit, Jens Hoppe, Hermann Nicolai; pp. 563 of: On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545-581 [doi:10.1016/0550-3213(88)90116-2, spire:261702, pdf, pdf]

• Emmanuel G. Floratos, John Iliopoulos, G. Tiktopoulos: A note on $SU(\infty)$ classical Yang-Mills theories, Physics Letters B 217 3 (1989) 285-288 [doi:10.1016/0370-2693(89)90867-8]

For the 2-torus:

• David B. Fairlie, Paul Fletcher, Cosmas K. Zachos: Trigonometric structure constants for new infinite-dimensional algebras, Physics Letters B 218 2 (1989) 203-206 [doi:10.1016/0370-2693(89)91418-4]

• Christopher N. Pope, Kellogg S. Stelle: $SU(\infty)$, $SU_+(\infty)$ and area-preserving algebras, Physics Letters B 226 3–4 (1989) 257-263 [doi:10.1016/0370-2693(89)91191-X]

• Hermann Nicolai, Robert C. Helling; p. 20 of: Supermembranes and M(atrix) Theory, In: Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry (1998) 29-74 [arXiv:hep-th/9809103, spire:476366]

Warning that the analogous statements for the Lie groups (as opposed to their Lie algebras) fail dramatically, for basic topological reasons:

• John Swain: On the limiting procedure by which $SDiff(T^2)$ and $SU(\infty)$ are associated [arXiv:hep-th/0405002]

• John Swain: The Topology of $SU(\infty)$ and the Group of Area-Preserving Diffeomorphisms of a Compact 2-manifold [arXiv:hep-th/0405003]

• John Swain: The Majorana representation of spins and the relation between $SU(\infty)$ and $SDiff(S^2)$ [arXiv:hep-th/0405004]

Analogous discussions:

for the tetrahedron:

• A. Wolski, J. S. Dowker: Area‐preserving diffeomorphisms of the tetrahedron, J. Math. Phys. 32 (1991) 857–863 [doi:10.1063/1.529343]

for $\mathfrak{sl}(n;\mathbb{N})$ and $\mathfrak{su}(n,n)$:

• William Donnelly, Laurent Freidel, Seyed Faroogh Moosavian, Antony J. Speranza: Matrix quantization of gravitational edge modes, J. High Energ. Phys. 2023, 163 (2023) [doi:10.1007/JHEP05(2023)163]

and in relation to the KP hierarchy:

• Kanehisa Takasaki, Takashi Takebe: $SDiff(2)$ KP hierarchy, Int. J. Mod. Phys. A 7 supp01B (1992) 889-922 [doi:10.1142/S0217751X92004099, arXiv:hep-th/9112046]