Borel subalgebra#Borel subalgebra relative to a base of a root system

Let $\backslash mathfrak\; g\; =\; \backslash mathfrak\{gl\}(V)$ 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 $\backslash mathfrak\; g$ amounts to specify a flag of V; given a flag $V\; =\; V\_0$

\supset V_1 \supset \cdots \supset V_n = 0, the subspace $\backslash mathfrak\; b\; =\; \backslash \{\; x\; \backslash in\; \backslash mathfrak\; g\; \backslash mid\; x(V\_i)\; \backslash subset\; V\_i,\; 1\; \backslash le\; i\; \backslash le\; n\; \backslash \}$ is a Borel subalgebra,{{harvnb|Serre|2000|loc=Ch I, § 6.}} 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.

Let $\backslash mathfrak\; g$ be a complex semisimple Lie algebra, $\backslash mathfrak\; h$ a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then $\backslash mathfrak\; g$ has the decomposition $\backslash mathfrak\; g\; =\; \backslash mathfrak\; n^-\; \backslash oplus\; \backslash mathfrak\; h\; \backslash oplus\; \backslash mathfrak\; n^+$ where $\backslash mathfrak\; n^\{\backslash pm\}\; =\; \backslash sum\_\{\backslash alpha\; >\; 0\}\; \backslash mathfrak\{g\}\_\{\backslash pm\; \backslash alpha\}$. Then $\backslash mathfrak\; b\; =\; \backslash mathfrak\; h\; \backslash oplus\; \backslash mathfrak\; n^+$ is the Borel subalgebra relative to the above setup.{{harvnb|Serre|2000|loc=Ch VI, § 3.}} (It is solvable since the derived algebra $[\backslash mathfrak\; b,\; \backslash mathfrak\; b]$ is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.{{harvnb|Serre|2000|loc=Ch. VI, § 3. Theorem 5.}})

Given a $\backslash mathfrak\; g$-module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for $\backslash mathfrak\; h$ and that (2) is annihilated by $\backslash mathfrak\{n\}^+$. It is the same thing as a $\backslash mathfrak\; b$-weight vector (Proof: if $h\; \backslash in\; \backslash mathfrak\; h$ and $e\; \backslash in\; \backslash mathfrak\{n\}^+$ with $[h,\; e]\; =\; 2e$ and if $\backslash mathfrak\{b\}\; \backslash cdot\; v$ is a line, then $0\; =\; [h,\; e]\; \backslash cdot\; v\; =\; 2\; e\; \backslash cdot\; v$.)

• Borel subgroup
• Parabolic Lie algebra

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• {{citation |first1=Neil |last1=Chriss |first2=Victor |last2=Ginzburg |title=Representation Theory and Complex Geometry |url=https://books.google.com/books?id=OZlCAAAAQBAJ&pg=PP1 |date=2009 |orig-year=1997 |publisher=Springer |isbn=978-0-8176-4938-8 }}.
• {{Citation | last1=Humphreys | first1=James E. | title=Introduction to Lie Algebras and Representation Theory | publisher=Springer-Verlag | isbn=978-0-387-90053-7 | year=1972 | url-access=registration | url=https://archive.org/details/introductiontoli00jame/page/83|ref={{harvid|Humphreys}}}}.
• {{Citation |url=https://books.google.com/books?id=7AHsSUrooSsC&pg=PA3|title=Algèbres de Lie semi-simples complexes|last=Serre|first=Jean-Pierre|date=2000|publisher=Springer|trans-title=Complex Semisimple Lie Algebras|isbn=978-3-540-67827-4|language=en|translator-last=Jones|translator-first=G. A.}}.

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Category:Representation theory

Category:Lie algebras

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