Cartan criterion

The Cartan criterion , which goes back to Élie Cartan , is a mathematical theorem from the theory of Lie algebras , which represents a criterion for the solvability of a Lie algebra. The resulting criterion for semi-simplicity is often also called the Cartan criterion . Some authors therefore speak more precisely of the Cartan criterion for solvability and the Cartan criterion for semi-simplicity .

Definitions

For a finite-dimensional vector space denote the trace of the endomorphism on and the general linear Lie algebra above , which is the Lie algebra of all endomorphisms with the commutator bracket . ${\displaystyle V}$ ${\displaystyle \mathrm {Spur} (x)}$ ${\displaystyle x}$ ${\displaystyle V}$ ${\displaystyle {\mathfrak {gl}}(V)}$ ${\displaystyle V}$

For a Lie algebra denote the Lie subalgebra of generated by all commutators . You can iterate that by defining. A Lie algebra is known to be called solvable, if there is one with . Finally, let it be the adjoint representation that maps each to the endomorphism . ${\displaystyle L}$ ${\displaystyle L^{(1)}:=[L,L]}$ ${\displaystyle [x,y],\,x,y\in L}$ ${\displaystyle L}$ ${\displaystyle L^{(n+1)}:=[L^{(n)},L^{(n)}]}$ ${\displaystyle L}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle L^{(n)}=\{0\}}$ ${\displaystyle \mathrm {ad} :L\rightarrow {\mathfrak {gl}}(L)}$ ${\displaystyle x\in L}$ ${\displaystyle \mathrm {ad} \,x:L\rightarrow L,\,y\mapsto [x,y]}$

Cartan criterion for solvability

Let it be a finite-dimensional vector space over a field of characteristic 0 and a Lie subalgebra of . Then the following statements are equivalent: ${\displaystyle V}$ ${\displaystyle L}$ ${\displaystyle {\mathfrak {gl}}(V)}$

• ${\displaystyle L}$ is resolvable.
• ${\displaystyle \mathrm {Spur} (xy)=0}$for everyone and . ${\displaystyle x\in [L,L]}$ ${\displaystyle y\in L}$

Corollary to the sentence

For a finite-dimensional Lie algebra over a field of characteristic 0, the following statements are equivalent: ${\displaystyle L}$

• ${\displaystyle L}$ is resolvable.
• ${\displaystyle \mathrm {Spur} ((\mathrm {ad} \,x)(\mathrm {ad} \,y))=0}$for everyone and . ${\displaystyle x\in [L,L]}$ ${\displaystyle y\in L}$

If namely is solvable, so is the homomorphic picture and because of this the mentioned condition follows from the above sentence. Conversely, if the condition is met, it follows from the above sentence that is solvable. Since the core of the adjoint representation is the center of the Lie algebra and this can be solved in a trivial way as an Abelian Lie algebra , the total solvability of follows . ${\displaystyle L}$ ${\displaystyle \mathrm {ad} (L)}$ ${\displaystyle \mathrm {ad} ([L,L])=[\mathrm {ad} (L),\mathrm {ad} (L)]}$ ${\displaystyle \mathrm {ad} (L)\subset {\mathfrak {gl}}(L)}$ ${\displaystyle L}$

If the definition of the killing form is used , the condition in the above corollary can also be written briefly as . ${\displaystyle \kappa (x,y):=\mathrm {Spur} ((\mathrm {ad} \,x)(\mathrm {ad} \,y))=0}$ ${\displaystyle \kappa ([L,L],L)=0}$

Cartan criterion for semi-simplicity

For a finite-dimensional Lie algebra over a field of characteristic 0, the following statements are equivalent: ${\displaystyle L}$

• ${\displaystyle L}$is semi-easy .
• The killing form on is not degenerate . ${\displaystyle L}$

Remarks

The Cartan criterion for solvability presented above is used to prove the criterion for semi-simplicity , which also goes back to Cartan, but it is not so called by all authors. The textbooks by Humphreys or Hilgert-Neeb mentioned below simply call the criterion for solvability the Cartan criterion and do not explicitly attribute the semi-simplicity criterion to Cartan, while the authors Sagle-Walde or Knapp, for example, use the sentence designations presented here.

The criteria do not apply in the case of positive characteristics of the base body. Is for a prime number and the Witt algebra with canonical base and product , then is a simple Lie algebra, the killing form of which is 0. This is a counterexample for both criteria: If the Cartan criterion for solvability were correct here, the condition would be trivially fulfilled due to the disappearance of the killing form and the algebra would have to be solvable, but it is simple. If the Cartan criterion for semi-simplicity were valid here, the killing form of the simple and thus semi-simple algebra would have to be non-degenerate, but it disappears identically. ${\displaystyle K=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle p\geq 5}$ ${\displaystyle W}$ ${\displaystyle K^{p}}$ ${\displaystyle e_{n},n\in K}$ ${\displaystyle [e_{i},e_{j}]:=(i-j)e_{i+j},\,i,j\in K}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle W}$

Individual evidence

1. ^ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3
2. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory , Berlin, New York: Springer-Verlag (1972), ISBN 978-0-387-90053-7 , Chapter II, 4.3: Cartan's Criterion
3. ^ Arthur A. Sagle, Ralph Walde: Introduction to Lie groups and Lie algebras , Academic Press (1973), ISBN 0-080-87366-9 , sentence 12.16
4. ^ Anthony W. Knapp: Lie Groups Beyond an Introduction , Birkhäuser (2002), ISBN 0-8176-4259-5 , sentence 1.46.
5. Jump up ↑ Arthur A. Sagle, Ralph Walde: Introduction to Lie groups and Lie algebras , Academic Press (1973), ISBN 0-080-87366-9 , sentence 12.17
6. ^ Anthony W. Knapp: Lie Groups Beyond an Introduction , Birkhäuser (2002), ISBN 0-8176-4259-5 , sentence 1.45
7. ↑ Dmitriy Rumynin: Modular Lie Algebras , Lecture Notes 2010