- #1

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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

- Thread starter Lapidus
- Start date

- #1

- 343

- 11

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

- #2

- 160

- 2

su(2) is a vector space over R with three generators; the general element of su(2) is a

sl(2) is a vector space over C with three generators; the general element of sl(2) is a

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.

- #3

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thanks Henry!!

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