Page "Unitary group" Paragraph 34

From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group ( reversing the Dynkin diagram A < sub > n </ sub >, which corresponds to transpose inverse ) and the field automorphism of the extension C / R ( namely complex conjugation ).

Both these automorphisms are automorphisms of the algebraic group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group.

The classical unitary group is a real form of this group, corresponding to the standard Hermitian form Ψ, which is positive definite.