If I define a Lie groupe as an analytical manifold of dimension N with the following group operations: $g(a) g(b) = g(p(a,b))$, $g^-1(a) = g(r(a))$ Wenn labelling elements of the group $a=(a_1, …, a_N), b=(b_1, …, b_N)$, with p and r analytic,
I am then not really satisfied with the standard SU(2) definition using matrices: for me this is a representation of the group, not the abstract group itself, which would be expressed as above (as what’s important is the structure of composition).
Using the properties of the 2x2 complex matrices of SU(2) I can derive $SU(2) = \{(a, b) \in \mathbb{C}^2 : a^2 + b^2 = 1; p((a,b), (c,d)) = (a c - \bar{b}d, \bar{a} d + b c); r((a,b)) = (\bar{a}, -b)$
Would this be a valid definition of the group SU(2)?
