I cannot quite vouch for exhaustive panoramas, but the crucial point is that *GL(N), SU(N)* matrices are representable in a nonhermitean basis discovered by Sylvester in 1882, the clock and shift matrices which he called nonions for *N*=3 (*long* before the Gell-Mann basis!), sedenions, etc. Their braiding relations, and maximal grading, and hence commutators, structure constants, and Casimirs!, are thus analytic in *N* and hence handily amenable to the *N⟶∞* limit.

They undergird a discrete truncation of the Heisenberg group explored by Weyl in 1927, but that is almost besides the point, except for the fact that, in a toroidal phase space, they can be organized to SU(N) generators with two integer indices, cf. a quarter-century old talk of mine which also does the SU(∞) gauge theory.

That is to say, the Moyal algebra on a toroidal phase space amounts to *SU(N)*, and the classical, ħ⟶0, limit of that algebra, which is the Poisson-Bracket algebra (an observation first made on the sphere by Hoppe, but made manifest on the torus here), is thus a Fourier-transform description of SU(∞).

The above summary talk bird-eye-views an expansive panoramas implicit in here ; and here ;and ;
available at ; and ; [and finally]
(https://www.academia.edu/6688013/Infinite-Dimensional_Algebras_and_a_Trigonometric_Basis_for_the_Classical_Lie_Algebras) with apologies for the massive document dump.

Now the Poisson-bracket algebra describes area-preserving diffeomorphisms on a notional phase space, and as such it lends itself to connecting this SU(∞) to the null string of Schild, basically with the Nambu action squared. Intriguingly, it has found applications in 2d hydrodynamics and the systematic study of Casimirs, also in use in large-N models in QFT, and the predictable supersymmetrization of such.

This post imported from StackExchange Physics at 2016-04-05 08:50 (UTC), posted by SE-user Cosmas Zachos