Lie algebra of an algebraic group

l0584801.png ~/encyclopedia/old_files/data/L058/L.0508480 104 0 104 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 $ G $ is the tangent space to $ G $ 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 $ G $ . The precise definition is as follows.

Let $ K $ be an algebraically closed field, $ G $ an affine algebraic $ K $ - group, $ A = K [G] $ the algebra of regular functions on $ G $ , and $ \mathop{\rm Lie}\nolimits (G) $ the set of all derivations of the $ K $ - algebra $ A $ that commute with automorphisms of $ A $ determined by left translations of $ G $ . The space $ \mathop{\rm Lie}\nolimits (G) $ is a Lie algebra with the operation $ [ D _{1} ,\ D _{2} ] = D _{1} \circ D _{2} - D _{2} \circ D _{1} $ ( see Lie algebra, linear), and the operation $ D ^{[p]} = D \circ \dots \circ D $ ( $ p $ factors) defines on $ \mathop{\rm Lie}\nolimits (G) $ a Lie $ p $ - algebra structure ($ p $ is equal to the characteristic of $ K $ if the latter is positive, and equal to 1 if the latter is zero). Let $ L (G) $ be the tangent space to $ G $ at the indentity $ e $ , that is, the vector space over $ K $ of all $ K $ - derivations from $ A $ to the $ A $ - module $ A / \mathfrak m _{e} $ , where $ \mathfrak m _{e} $ is the maximal ideal of $ e $ , and let $ \phi _{e} : \ A \rightarrow A / \mathfrak m _{e} $ be the canonical homomorphism. For any $ D \in \mathop{\rm Lie}\nolimits (G) $ the composition $ \phi _{e} \circ D $ is an element of $ L (G) $ , and the mapping $ \mathop{\rm Lie}\nolimits (G) \rightarrow L (G) $ defined by the formula $ D \rightarrow \phi _{e} \circ D $ is an isomorphism of vector spaces over $ K $ . This makes it possible to carry over the structure of a Lie $ p $ - algebra from $ \mathop{\rm Lie}\nolimits (G) $ to $ L (G) $ . This Lie $ p $ - algebra $ L (G) $ is called the Lie algebra of the algebraic group $ G $ . If $ k $ is a subfield of $ K $ and if $ G $ is defined over $ k $ , then the left-invariant $ k $ - derivations of the $ k $ - algebra $ A _{k} \subset A $ that define the $ k $ - structure on $ G $ form a $ k $ - structure on $ \mathop{\rm Lie}\nolimits (G) $ , and the isomorphism mentioned above is defined over $ k $ .

Example. Let $ V $ be a finite-dimensional vector space over $ K $ and let $ G = \mathop{\rm GL}\nolimits (V) $ be the algebraic group of all automorphisms of $ V $ . Then the tangent space to $ G $ at $ e $ is naturally identified with the vector space $ \mathop{\rm End}\nolimits \ V $ of all endomorphisms of $ V $ , and the structure of a Lie algebra of the algebraic group $ G $ on $ \mathop{\rm End}\nolimits \ V $ is specified by the formulas $ [ X ,\ Y ] = XY - YX $ , $ X ^{[p]} = X ^{p} $ . The resulting Lie algebra is denoted by $ \mathfrak g \mathfrak l (V) $ .

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 $ G $ is equal to the dimension of $ G $ . The Lie algebras of an algebraic group $ G $ 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 $ H $ is an algebraic subgroup of an algebraic group $ G $ , then $ L (H) $ is a subalgebra of $ L (G) $ . Moreover, let $ J $ be the ideal of all regular functions on $ G $ that vanish on $ H $ . Then, identifying $ L (G) $ with $ \mathop{\rm Lie}\nolimits (G) $ , one can describe $ L (H) $ as the set of all elements of $ \mathop{\rm Lie}\nolimits (G) $ that annihilate $ J $ . This description is particularly convenient for the examination of linear algebraic groups, that is, algebraic subgroups $ G $ of $ \mathop{\rm GL}\nolimits (V) $ . Namely, let $ J $ be the ideal of $ K [ \mathop{\rm End}\nolimits \ V ] $ consisting of elements equal to zero on $ G $ . Then $ L (G) \subset \mathfrak g \mathfrak l (V) $ consists precisely of endomorphisms $ X $ of $ V $ such that the derivation of the algebra $ K [ \mathop{\rm End}\nolimits \ V ] $ induced by the endomorphism $ Y \mapsto X Y $ of $ \mathop{\rm End}\nolimits \ V $ takes $ J $ into itself. The operations in $ L (G) $ are induced by the operations in $ \mathfrak g \mathfrak l (V) $ described above.

If $ p = 1 $ , 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 $ \mathop{\rm GL}\nolimits (V) $ ) are distinguished among all Lie subalgebras of $ \mathfrak g \mathfrak l (V) $ by means of an intrinsic criterion (see Lie algebra, algebraic). In the case $ p > 1 $ 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 $ p=1 $ , 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 $ p = 2 $ ), etc.

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