Levi's theorem (Lie algebra)

The set of Levi , named after Eugenio Elia Levi , is a set from the theory of Lie algebras from 1905, of the decomposition of a finite, real or complex Lie algebra in a semi direct sum of a half simple and a resolvable Lie Has algebra as its content; this is also called the Levi decomposition .

Terms

That with designated radical of a Lie algebra is the largest contained in it resolvable Ideal . The quotient after the radical has no radical, that is, the radical is the null space and by definition is semi-simple. ${\displaystyle \mathrm {rad} ({\mathfrak {g}})}$ ${\displaystyle {\mathfrak {g}}/\mathrm {rad} ({\mathfrak {g}})}$

Formulation of the sentence

Let it be a finite-dimensional real or complex Lie algebra. Then there is a semi-simple Lie algebra with ( vector space sum ). ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {s}}\subset {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus \mathrm {rad} ({\mathfrak {g}})}$

Since is an ideal, it is even a semidirect sum of Lie algebras. ${\displaystyle \mathrm {rad} ({\mathfrak {g}})\subset {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {s}}\ltimes \mathrm {rad} ({\mathfrak {g}})}$

Line of evidence

Since is an ideal, one gets a short exact sequence ${\displaystyle \mathrm {rad} ({\mathfrak {g}})\subset {\mathfrak {g}}}$

${\displaystyle 0\rightarrow \mathrm {rad} ({\mathfrak {g}})\rightarrow {\mathfrak {g}}\,{\xrightarrow {\varphi }}\,{\mathfrak {g}}/\mathrm {rad} ({\mathfrak {g}})\rightarrow 0}$.

The above-mentioned decomposition into a vector space sum results immediately when the decay of this sequence is shown, that is, the existence of a Lie algebra homomorphism , so that the identical mapping is on . then do what is requested. This is exactly the content of the following theorem, which can be proved with the help of Whitehead's 2nd lemma : ${\displaystyle \psi :{\mathfrak {g}}/\mathrm {rad} ({\mathfrak {g}})\rightarrow {\mathfrak {g}}}$ ${\displaystyle \varphi \circ \psi }$ ${\displaystyle {\mathfrak {g}}/\mathrm {rad} ({\mathfrak {g}})}$ ${\displaystyle {\mathfrak {s}}:=\psi ({\mathfrak {g}}/\mathrm {rad} ({\mathfrak {g}}))}$

Let it be a finite-dimensional real or complex Lie algebra. Then the short exact sequence falls apart ${\displaystyle {\mathfrak {g}}}$

${\displaystyle 0\rightarrow \mathrm {rad} ({\mathfrak {g}})\rightarrow {\mathfrak {g}}\,{\xrightarrow {\varphi }}\,{\mathfrak {g}}/\mathrm {rad} ({\mathfrak {g}})\rightarrow 0}$.

Levi's theorem is a simple corollary of this theorem.

Levi complement

A semi-simple Lie algebra is called Levi's complement if the direct vector space sum with the radical is whole . Therefore, Levi's theorem can be briefly formulated as follows: ${\displaystyle {\mathfrak {s}}\subset {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$

Every finite-dimensional real or complex Lie algebra has a Levi complement.

See also

• Levi decomposition (Lie group)

Individual evidence

1. ^ EE Levi: "Sulla struttura dei gruppi finiti e continui", Atti della Reale Accademia delle Scienze di Torino, Volume XL: pages 551-565
2. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , sentence II.4.7
3. ^ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , Corollary II.4.8