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[itex]\mathbb{R}^3[/itex] has an associative multiplication [itex] \mu:\mathbb{R}^3\times \mathbb{R}^3 \rightarrow \mathbb{R}^3[/itex] given by [tex] \mu((x,y,z),(x',y',z'))=(x+x', y+y', z+z'+xy'-yx')[/tex]
Determine an identity and inverse so that this forms a Lie group.
Well, clearly e=(0,0,0) and the inverse element is (-x,-y,-z)
Pick a basis for the Lie algebra of this Lie group and calculate their commutators, obtaining the structure constants of the Lie algebra.
This is the part I'm having trouble with. I'm not really sure where to start! Any help would be much appreciated!
Determine an identity and inverse so that this forms a Lie group.
Well, clearly e=(0,0,0) and the inverse element is (-x,-y,-z)
Pick a basis for the Lie algebra of this Lie group and calculate their commutators, obtaining the structure constants of the Lie algebra.
This is the part I'm having trouble with. I'm not really sure where to start! Any help would be much appreciated!