Constructing algebraic groups from their Lie algebras. (English) Zbl 1180.20038

de Graaf, Willem A.

J. Symb. Comput. 44, No. 9, 1223-1233 (2009).

Let \(F\) be a field of characteristic \(0\), with algebraic closure \(K\). A connected algebraic group \(G\) over \(F\) is uniquely determined by its Lie algebra \(\mathfrak g\).
Given an algebraic Lie algebra \(\mathfrak g\) over \(F\) (i.e., one which is the Lie algebra of an algebraic group over \(F\)), the author gives an algorithm for computing the unique connected algebraic group \(G\) with Lie algebra \(\mathfrak g\). Here, \(\mathfrak g\) is given as a set of basis vectors inside \(\mathfrak{gl}(n,F)\) for some \(n\). The algorithm then returns a set of defining polynomials for the group \(G\). The algorithm may be implemented in a computer algebra system, such as MAGMA.
The algorithm consists of a number of subalgorithms. The first step is to decompose \(\mathfrak g\) into a semisimple part \(\mathfrak l\), a toral subalgebra \(\mathfrak t\), and an ideal \(\mathfrak n\) consisting of nilpotent matrices. Then algorithms are given to construct algebraic groups corresponding to a reductive, a semisimple, a toral, or a nilpotent Lie algebra. The reductive group is not necessarily connected; for the user who does not require connectedness, this algorithm produces a group corresponding to a reductive Lie algebra directly. Finally, the author gives an algorithm for putting the groups corresponding to these subalgebras together to obtain (defining polynomial equations for) the (connected) algebraic group corresponding to an arbitrary algebraic Lie algebra \(\mathfrak g\). In the last section, he describes some of his experiences with the implementation, in particular, the practical performance of the algorithm.

Reviewer: Annegret Paul (Kalamazoo)

MSC:

20G15	Linear algebraic groups over arbitrary fields
17B45	Lie algebras of linear algebraic groups
17-04	Software, source code, etc. for problems pertaining to nonassociative rings and algebras
20-04	Software, source code, etc. for problems pertaining to group theory
17-08	Computational methods for problems pertaining to nonassociative rings and algebras
68W30	Symbolic computation and algebraic computation

Keywords:

algebraic groups; algebraic Lie algebras; maximal reductive subgroups; computational methods

Software:

Magma; SINGULAR

References:

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