In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra is a maximal solvable subalgebra. The notion is named after Armand Borel.
If the Lie algebra is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup.
Borel subalgebra associated to a flag
Let be the Lie algebra of the endomorphisms of a finite-dimensional vector space V over the complex numbers. Then to specify a Borel subalgebra of amounts to specify a flag of V; given a flag , the subspace is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of V.
Borel subalgebra relative to a base of a root system
Let be a complex semisimple Lie algebra, a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then has the decomposition where . Then is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.)
Given a -module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for and that (2) is annihilated by . It is the same thing as a -weight vector (Proof: if and with and if is a line, then .)
See also
Borel subgroup
Parabolic Lie algebra
References
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