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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].
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