Weyl's theorem (Lie algebra)

The set of Weyl , named after Hermann Weyl is an important sentence from the theory of Lie algebras . It basically means that finite-dimensional representations of semisimple Lie algebras can be composed of irreducible ones , provided that the basic field is algebraically closed and has the characteristic 0.

Terms

A representation of a Lie algebra over a vector space is a Lie algebra homomorphism of , the latter denoting the general linear Lie algebra over , i.e. H. the set of all linear operators with the commutator bracket as a link. The dimension of the vector space is also called the dimension of the representation. A subspace is called invariant if for all . Invariant subspaces are interesting because there is again a representation. One always has the so-called trivial invariant subspaces and ; if there is only this, the representation is called irreducible, because it cannot be simplified (reduced) by further invariant subspaces. A representation is now called completely reducible if there are invariant subspaces that differ from the zero vector space , so that is the direct sum of these subspaces and every representation is irreducible. Completely reducible representations can therefore be broken down into their irreducible components. Therefore theorems that ensure the complete reducibility of representations are very important, especially when one knows all irreducible representations. ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle \pi :L\rightarrow \mathrm {gl} (V)}$ ${\displaystyle V}$ ${\displaystyle W\subset V}$ ${\displaystyle \pi (x)w\in W}$ ${\displaystyle x\in L,w\in W}$ ${\displaystyle \pi _{W}:L\rightarrow \mathrm {gl} (W),\pi _{W}(x):=\pi (x)|_{W}}$ ${\displaystyle W=\{0\}}$ ${\displaystyle W=V}$ ${\displaystyle \pi :L\rightarrow \mathrm {gl} (V)}$ ${\displaystyle W_{1},\ldots ,W_{n}}$ ${\displaystyle V=W_{1}\oplus \ldots \oplus W_{n}}$ ${\displaystyle \pi _{W_{i}}}$

Formulation of the sentence

Every finite-dimensional representation of a semi-simple Lie algebra over an algebraically closed field of characteristic 0 is completely reducible.

A proof for -Lie algebras can be found in the textbook by Hilgert and Neeb given below, where this theorem is reduced to Whitehead's so-called lemma about the disappearance of certain cohomology groups . One proof that avoids cohomology theory is found in Humphreys. ${\displaystyle \mathbb {C} }$

Positive characteristic

Weyl's theorem becomes wrong for characteristic . If there is such a field, consider Lie algebra ${\displaystyle p>0}$ ${\displaystyle \mathbb {K} }$

${\displaystyle L=\mathrm {sl} _{2}(\mathbb {K} )}$,

known from

${\displaystyle e={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad f={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad h={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

is produced. For this Lie algebra is simple, especially semi-simple. ${\displaystyle p\geq 3}$

Furthermore, let the vector space of the polynomials be in two indeterminates. Through the formulas ${\displaystyle V:=\mathbb {K} [x,y]}$

${\displaystyle \pi (e)(v):=x\cdot {\frac {\partial v}{\partial y}},\quad \pi (f)(v):=y\cdot {\frac {\partial v}{\partial x}},\quad \pi (h)(v):=x\cdot {\frac {\partial v}{\partial x}}-y\cdot {\frac {\partial v}{\partial y}}}$    (for ) ${\displaystyle v\in V}$

an infinite-dimensional representation is defined. Since the operation defined in this way leaves the degrees of the polynomials unchanged, the subspaces generated by the homogeneous polynomials are invariant. One can now show that the finite-dimensional representation is not completely reducible. In fact, the subspace generated by and is invariant, because it is the zero representation, since the derivatives generate the factor , which equates to the characteristic of multiplication by 0 for fields, and there are no invariant, direct summands of in . Hence Weyl's theorem is not valid here. ${\displaystyle \pi :L\rightarrow \mathrm {gl} (V)}$ ${\displaystyle L}$ ${\displaystyle x^{k}y^{n-k},\,k=0,\ldots n}$ ${\displaystyle V_{n}}$ ${\displaystyle \pi _{V_{p}}:L\rightarrow \mathrm {gl} (V_{p})}$ ${\displaystyle x^{p}}$ ${\displaystyle y^{p}}$ ${\displaystyle W\subset V_{p}}$ ${\displaystyle \pi _{W}}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle W}$ ${\displaystyle V_{p}}$

See also

• Maschke's theorem for an analogous situation in group theory .

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 , Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Section 6.3: Weyl's Theorem
3. ↑ Dimitriy Rumynin: Modular Lie-Algebras , Chapter 2: Things that fail in positive characteristic