Representations of the Lie Algebra of SL(2, ℝ)

Abstract

We begin with a study of representations of sl(2, R). For simplicity of notation, we will write sl(2) for sl(2, R). Recall Example 1.1.8 in Chapter I where we introduced a basis {h, e⁺, e^-} for sl(2):

$$ h = \left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & { - 1} \\ \end{array} } \right],\,{e^{ + }} = \left[ {\begin{array}{*{20}{c}} 0 & 1 \\ 0 & 0 \\ \end{array} } \right],\,{e^{ - }} = \left[ {\begin{array}{*{20}{c}} 0 & 0 \\ 1 & 0 \\ \end{array} } \right] $$

((0.0.1))

with the commutation relations:

$$ \left[ {h,{e^{ + }}} \right] = 2{e^{ + }},\,\left[ {h,{e^{ - }}} \right] = - 2{e^{ - }},\,\left[ {{e^{ + }},{e^{ - }}} \right] = h $$

((0.0.2))

.