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Lie Groups

Groups

We start with defining points in \(\mathbb{R}^3 \). These are just what you might three numbers, \( \boldsymbol{p}=(p_1, p_2, p_3) \), that describe a location in \(\mathbb{R}^3 \). You can add any two points \[ \boldsymbol{a}+\boldsymbol{b}=(a_1+b_1,a_2+b_2,a_3+b_3) \] You can also multiply by a scalar, a real number, \( c \in \mathbb{R} \). \[ c\boldsymbol{p}=(cp_1, cp_2, cp_3) \] This space of points is isomorphic to what we usually would call a three dimensional vector space, but we'll follow O'Neill and be careful saying what is a vector and what is a point because soon we will have the notion of a vector being described by two connecting points.