- Linear algebraic group
In
mathematics , a linear algebraic group is asubgroup of the group of invertible "n"×"n" matrices (undermatrix multiplication ) that is defined bypolynomial equations. An example is theorthogonal group , defined by the relation MTM = I where MT is thetranspose of M.The main examples of linear algebraic groups are certain of the
Lie group s, where the underlying field is the real or complex field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by thePeter-Weyl theorem .)These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered byÉlie Cartan ,Ludwig Maurer ,Wilhelm Killing , andSophus Lie in the 1880s and 1890s in the context ofdifferential equation s andGalois theory . However, a purely algebraic theory wasn't sought for until around 1950, withArmand Borel as one of its pioneers. ThePicard-Vessiot theory did lead to algebraic groups.The first basic theorem of the subject is that any "affine"
algebraic group is a linear algebraic group: that is, anyaffine variety V that has an algebraic group law has a "faithful"linear representation , over the same field. For example the "additive group" of an "n"-dimensionalvector space has a faithful representation as "n"+1×"n"+1 matrices.One can define the
Lie algebra of an algebraic group purely algebraically (it consists of thedual number points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one Go containing the identity will be anormal subgroup of G.One of the first uses for the theory was to define the
Chevalley group s.The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of
Borel subgroup s B. These turn out to be maximal as connected solvable subgroups (i.e., subgroups withcomposition series having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type); and also minimal such that G/B is aprojective variety .Non-algebraic Lie groups
There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.
*Any Lie group with an infinite group of components G/Go cannot be realized as an algebraic group (see
identity component ).
*The center of a linear algebraic group is again a linear algebraic group. Thus, any group whose center has infinitely many components is not a linear algebraic group. An interesting example is theuniversal cover of SL2(R). This is a Lie group that maps infinite-to-one to SL2(R), since thefundamental group is hereinfinite cyclic - and in fact the cover has no faithful matrix representation.
*The generalsolvable Lie group need not have a group law expressible by polynomials.ee also
*
Differential Galois theory
*Group of Lie type is a group of rational points of a linear algebraic group.References
A good introduction to the theory of linear algebraic groups is:
*Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
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