In this chapter we describe techniques for computing with *-algebras, namely algebras equipped with an anti-automorphism x |-> x * of order at most 2 (an involution or star). For further information on involutions and the structure of *-algebras see [Alb61] and [KMRT98].
The principal application of these techniques is currently to isometry groups of systems of reflexive forms (and the intimately related study of intersections of classical groups). However, it is also possible to use the techniques to compute with group algebras of moderate dimension.
To any set of reflexive forms defined on a common vector space (a system of forms) one may associate a matrix *-algebra called the adjoint algebra of the system. The group of units of this adjoint algebra contains a natural subgroup of unitary elements, namely those elements x satisfying the condition x * =x - 1. The group of unitary elements coincides with the group of isometries of the system of forms, which is also the intersection of the general classical groups associated with these forms.
The StarAlgebras package provides functions that enable the user to investigate the structure of *-algebras. It also provides functions to compute and determine the structure of the group of isometries of a system of reflexive forms, and to compute intersections of arbitrary collections of classical groups defined on a common vector space.
The algorithms are mainly due to Peter Brooksbank and James Wilson [BW12a], [BW12b].