In mathematics, the Lie窶適olchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and
a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that
That is, マ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation. This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all .
It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. In fact, this is another way to state the Lie窶適olchin theorem.
The result for Lie algebras was proved by Sophus Lie ( 1876) and for algebraic groups was proved by Ellis Kolchin ( 1948, p.19).
The Borel fixed point theorem generalizes the Lie窶適olchin theorem.
Sometimes the theorem is also referred to as the Lie窶適olchin triangularization theorem because by induction it implies that with respect to a suitable basis of V the image has a triangular shape; in other words, the image group is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(n,K): the image is simultaneously triangularizable.
The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.
If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complex numbers of absolute value one is a one-dimensional commutative (and therefore solvable) linear algebraic group over the real numbers which has a two-dimensional representation into the special orthogonal group SO(2) without an invariant (real) line. Here the image of is the orthogonal matrix
In mathematics, the Lie窶適olchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and
a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that
That is, マ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation. This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all .
It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. In fact, this is another way to state the Lie窶適olchin theorem.
The result for Lie algebras was proved by Sophus Lie ( 1876) and for algebraic groups was proved by Ellis Kolchin ( 1948, p.19).
The Borel fixed point theorem generalizes the Lie窶適olchin theorem.
Sometimes the theorem is also referred to as the Lie窶適olchin triangularization theorem because by induction it implies that with respect to a suitable basis of V the image has a triangular shape; in other words, the image group is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(n,K): the image is simultaneously triangularizable.
The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.
If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complex numbers of absolute value one is a one-dimensional commutative (and therefore solvable) linear algebraic group over the real numbers which has a two-dimensional representation into the special orthogonal group SO(2) without an invariant (real) line. Here the image of is the orthogonal matrix