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An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion. Its tangent space at the identity carries a natural Lie algebra structure.

A Lie algebra over a field is a vector space $\mathfrak{g}$ with a bilinear operation $[x,y]$, called the Lie bracket, that is alternating and satisfies the Jacobi identity.

A Coxeter group is a group with a presentationwhere $m_{ii}=1$ and $m_{ij}=m_{ji}\in \{2,3,\dots ,\infty \}$ for $i\ne j$. The pair consisting of $W$ and its distinguished generating reflections is a Coxeter system.