Lie algebra of an algebraic group
The analogue of the Lie algebra of an analytic group, which relates to the case of affine algebraic groups. As in the analytic case, the Lie algebra of an algebraic group 
is the tangent space to 
at the identity, and the structure of a Lie algebra is defined on it by means of left-invariant derivations of the algebra of functions on 
. The precise definition is as follows.
Let 
be an algebraically closed field, 
an affine algebraic 
-group, 
the algebra of regular functions on 
, and 
the set of all derivations of the 
-algebra 
that commute with automorphisms of 
determined by left translations of 
. The space 
is a Lie algebra with the operation 
(see Lie algebra, linear), and the operation 
(
factors) defines on 
a Lie 
-algebra structure (
is equal to the characteristic of 
if the latter is positive, and equal to 1 if the latter is zero). Let 
be the tangent space to 
at the indentity 
, that is, the vector space over 
of all 
-derivations from 
to the 
-module 
, where 
is the maximal ideal of 
, and let 
be the canonical homomorphism. For any 
the composition 
is an element of 
, and the mapping 
defined by the formula 
is an isomorphism of vector spaces over 
. This makes it possible to carry over the structure of a Lie 
-algebra from 
to 
. This Lie 
-algebra 
is called the Lie algebra of the algebraic group 
. If 
is a subfield of 
and if 
is defined over 
, then the left-invariant 
-derivations of the 
-algebra 
that define the 
-structure on 
form a 
-structure on 
, and the isomorphism mentioned above is defined over 
.
Example. Let 
be a finite-dimensional vector space over 
and let 
be the algebraic group of all automorphisms of 
. Then the tangent space to 
at 
is naturally identified with the vector space 
of all endomorphisms of 
, and the structure of a Lie algebra of the algebraic group 
on 
is specified by the formulas 
, 
. The resulting Lie algebra is denoted by 
.
Lie algebras of algebraic groups have a number of properties analogous to those of Lie algebras of analytic groups. Thus, the differential of a homomorphism of algebraic groups at the identity is a homomorphism of their Lie algebras. The dimension of the Lie algebra of an algebraic group 
is equal to the dimension of 
. The Lie algebras of an algebraic group 
and of its connected component of the identity coincide. The differential of the adjoint representation of an algebraic group is the adjoint representation of its Lie algebra (cf. also Adjoint representation of a Lie group). If 
is an algebraic subgroup of an algebraic group 
, then 
is a subalgebra of 
. Moreover, let 
be the ideal of all regular functions on 
that vanish on 
. Then, identifying 
with 
, one can describe 
as the set of all elements of 
that annihilate 
. This description is particularly convenient for the examination of linear algebraic groups, that is, algebraic subgroups 
of 
. Namely, let 
be the ideal of 
consisting of elements equal to zero on 
. Then 
consists precisely of endomorphisms 
of 
such that the derivation of the algebra 
induced by the endomorphism 
of 
takes 
into itself. The operations in 
are induced by the operations in 
described above.
If 
, then the connection between affine algebraic groups and their Lie algebras is essentially as close as the connection between analytic groups and their Lie algebras. This makes it possible to reduce substantially the study of affine algebraic groups to the study of their Lie algebras and conversely. Moreover, Lie algebras of linear algebraic groups (that is, algebraic subgroups of 
) are distinguished among all Lie subalgebras of 
by means of an intrinsic criterion (see Lie algebra, algebraic). In the case 
this connection is not so close and substantially loses its significance. Namely, in this case in general only those results remain true that make it possible to extract from properties of a group information about properties of its Lie algebra. On the contrary, many theorems that, if 
, establish this connection in the reverse direction cease to be true. For example, there may exist various connected subgroups of a given group with coinciding Lie algebras; the Lie algebra of a non-solvable group may be solvable (this is so, for example, for the group of matrices of order 2 with determinant 1 for 
), etc.
References
| [1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
| [2] | C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) |
Comments
References
| [a1] | G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) |
Lie algebra of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra_of_an_algebraic_group&oldid=21785