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general linear group

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Definition

Given a field kk, the general linear group GL(n,k)GL(n,k) (or GL n(k)GL_n(k)) is the group of invertible linear transformations of the vector space k nk^n. It can be canonically identified with the group of n×nn\times n matrices with entries in kk having nonzero determinant.

This group can be considered as a (quasi-affine) subvariety of the affine space M n×n(k)M_{n\times n}(k) of square matrices of size nn defined by the condition that the determinant of a matrix is nonzero. It can be also presented as an affine subvariety of the affine space M n×n(k)×kM_{n \times n}(k) \times k defined by the equation det(M)t=1\det(M)t = 1 (where MM varies over the factor M n×n(k)M_{n \times n}(k) and tt over the factor kk).

This variety is an algebraic kk-group, and if kk is the field of real or complex numbers it is a Lie group over kk.

One can in fact consider the set of invertible matrices over an arbitrary unital ring, not necessarily commutative. Thus GL n:RGL n(R)GL_n: R\mapsto GL_n(R) becomes a presheaf of groups on Aff=Ring opAff=Ring^{op} where one can take rings either in commutative or in noncommutative sense. In the commutative case, this functor defines a group scheme; it is in fact an affine group scheme represented by the commutative ring R=[x 11,,x nn,t]/(det(X)t1)R = \mathbb{Z}[x_{11}, \ldots, x_{n n}, t]/(det(X)t - 1).

Coordinate rings of general linear groups and of special general linear groups have quantum deformations called quantum linear group?s.

Stable and unstable versions:

The above is sometimes referred to as the unstable general linear group, whilst the result if one lets nn go to infinity is called the stable general linear group of RR, and is then written GL(R)GL(R) with no suffix.

References

  • O.T. O’Meara, Lectures on Linear Groups, Amer. Math. Soc., Providence, RI, 1974.

  • B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

Revised on March 22, 2014 08:31:32 by Urs Schreiber (82.113.121.50)