Real vs. Complex: Understanding the Difference Between su(2) and sl(2) Algebras

[Lapidus]

one standard basis of su(2) are the 2x2 matrices (i 0;0 -i), (0 i; i 0), (0 1;-1 0)

whereas the standard basis of sl(2) are (1 ; 0 -1), (0 1; 0 0), (0 0;-1 0)

Why then is su(2) called a real algebra, but not sl(2)?

thanks

 

[henry_m]

The answer is the field over which the vector space is defined:
su(2) is a vector space over R with three generators; the general element of su(2) is a real linear combination of the generators.
sl(2) is a vector space over C with three generators; the general element of sl(2) is a complex linear combination of the generators. It's sometimes called sl(2,C) or similar; sl(2,R) would be a different Lie algebra.

Incidentally, the basis you have given for su(2) also does perfectly well as a basis for sl(2), but over C. sl(2) is the complexification of su(2).

From a mathematical point of view the algebra is defined abstractly, without any reference to a basis. The fact that there is a standard representation by matrices with complex or real entries has no bearing on whether the algebra is complex or real.

 

[Lapidus]

thanks Henry!