I am trying to understand Lie algebras and I've come across this $$\mathfrak{sl}(2,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}\cong\mathfrak{sl}(2,\mathbb{C})\cong\mathfrak{so}(3,\mathbb{C}),$$ but I don't really know how the first object works; does it just consist of tensoring each matrix entry by $i=\sqrt{-1}$? Or is it somewhat more complex?
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2either by $i$ or by $1.$ https://en.wikipedia.org/wiki/Complexification – Anne Bauval Feb 21 '23 at 16:56
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@AnneBauval so, as $\mathfrak{sl}(2,\mathbb{R})$ is generated by $h=(1,0,0,-1)$, $e=(0,1,0,0)$ and $f=(0,0,1,0)$ (as matrices) then will the tensor product be generated by $h\otimes 1,\ldots, f\otimes i$? – valkyriebel Feb 21 '23 at 17:05
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2Yes exactly, over $\mathbb{R}$ the elements $h\otimes 1,h\otimes i, e\otimes 1,e\otimes i,f\otimes 1,f\otimes i$ form a basis. – student91 Feb 21 '23 at 17:07
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Related: https://math.stackexchange.com/q/188791/96384 – Torsten Schoeneberg Feb 22 '23 at 00:48