Algebras

See also the subsection Representation Theory.

Fast algorithms for computing the Jacobson radical and unit group of a matrix algebra over a finite field were designed and implemented by Peter Brooksbank (Bucknell) and Eamonn O'Brien (Auckland).

A package for computing with algebras equipped with an involution (*-algebras) has been contributed by Peter Brooksbank (Bucknell) and James Wilson (Ohio State).

An algorithm designed and implemented by Jon Carlson and Graham Matthews (Athens, Ga.) provides an efficient means for constructing presentations for matrix algebras.

For matrix algebras defined over a finite field, Jon Carlson (Athens, Ga.) designed and implemented algorithms for the Jacobson radical and unit group which are faster than the Brooksbank–O'Brien algorithms for larger examples.

Markus Kirschmer (Aachen) has written a number of optimized routines for definite quaternion algebras (over number fields).

Markus Kirschmer has also contributed a package for quaternion algebras defined over the function fields Fq[t], for q odd. The package includes calculation of the normaliser of an order and an efficient algorithm for computing the two-sided ideal classes of an order in a definite quaternion algebra (over ℤ or Fq[t]).

Quaternion algebras over the rational field ℚ were originally implemented by David Kohel (Singapore–NUS, Magma).

The vector enumeration program of Steve Linton (St. Andrews) provides an alternative to the use of Gröbner basis techniques for constructing a matrix representation of a finitely presented associative algebra.

Constructions for various types of non-associative algebras have been contributed by Josh Maglione and James Wilson (both Colorado State U) as part of their Multilinear Algebra package.

A package for computing with Clifford algebras defined over finite fields was developed by Don Taylor (Sydney) and released in Magma V2.18 (2011).

John Voight (Vermont) produced the package for quaternion algebras over number fields.