The finite groups of matrices with coefficients in Q generated by reflections, known as Weyl groups, are a fundamental building block in the classification of semisimple complex Lie algebras and Lie groups, as well as semisimple algebraic groups over arbitrary algebraically closed fields. They are also a foundation for many other significant mathematical theories, including braid groups and Hecke algebras. The Weyl groups are particular cases of complex reflection groups, finite groups of matrices with coefficients in a finite abelian extension of Q generated by “pseudo-reflections” (elements whose vector space of fixed points is a hyperplane) — if the coefficients belong to R, then these are the finite Coxeter groups.