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Radical of a Lie algebra

In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak{g} is the largest solvable ideal of \mathfrak{g}.{{citation

| last1 = Hazewinkel | first1 = Michiel

| last2 = Gubareni | first2 = Nadiya

| last3 = Kirichenko | first3 = V. V.

| doi = 10.1090/surv/168

| isbn = 978-0-8218-5262-0

| location = Providence, RI

| mr = 2724822

| page = 15

| publisher = American Mathematical Society

| series = Mathematical Surveys and Monographs

| title = Algebras, Rings and Modules: Lie Algebras and Hopf Algebras

| url = https://books.google.com/books?id=Q5K3vREGVhAC&pg=PA15

| volume = 168

| year = 2010}}.

The radical, denoted by {\rm rad}(\mathfrak{g}), fits into the exact sequence

:0 \to {\rm rad}(\mathfrak{g}) \to \mathfrak g \to \mathfrak{g}/{\rm rad}(\mathfrak{g}) \to 0.

where \mathfrak{g}/{\rm rad}(\mathfrak{g}) is semisimple. When the ground field has characteristic zero and \mathfrak g has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of \mathfrak g that is isomorphic to the semisimple quotient \mathfrak{g}/{\rm rad}(\mathfrak{g}) via the restriction of the quotient map \mathfrak g \to \mathfrak{g}/{\rm rad}(\mathfrak{g}).

A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.

Definition

Let k be a field and let \mathfrak{g} be a finite-dimensional Lie algebra over k. There exists a unique maximal solvable ideal, called the radical, for the following reason.

Firstly let \mathfrak{a} and \mathfrak{b} be two solvable ideals of \mathfrak{g}. Then \mathfrak{a}+\mathfrak{b} is again an ideal of \mathfrak{g}, and it is solvable because it is an extension of (\mathfrak{a}+\mathfrak{b})/\mathfrak{a}\simeq\mathfrak{b}/(\mathfrak{a}\cap\mathfrak{b}) by \mathfrak{a}. Now consider the sum of all the solvable ideals of \mathfrak{g}. It is nonempty since \{0\} is a solvable ideal, and it is a solvable ideal by the sum property just derived. Clearly it is the unique maximal solvable ideal.

Related concepts

  • A Lie algebra is semisimple if and only if its radical is 0.
  • A Lie algebra is reductive if and only if its radical equals its center.

See also

References

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Category:Lie algebras