Lie Groups/Algebras, Notes 2: Representations and Types

Recall a representation of an algebra A on an R-module M is an algebra morphism \rho:A\to End_R(M).  When we have a Lie algebra \mathfrak{g}, a representation of it on an R-module M is an algebra morphism

\rho:\mathfrak{g}\to L(End_R(M)).

Hence in particular we have \rho(xy)(m)=\left(\rho(x)\rho(y)-\rho(y)\rho(x)\right)(m).  We correspondingly call M a \mathfrak{g}module.  A representation is faithful if it is injective.  It is simple/irreducible if its \mathfrak{g}-module is simple.

Let \rho_1 and \rho_2 be representations of Lie algebras \mathfrak{g}_1 and \mathfrak{g}_2 on M_1 and M_2 respectively.  Then \rho_1\otimes\rho_2 is a representation of \mathfrak{g}_1\times\mathfrak{g}_2 on M_1\otimes M_2 defined by

\begin{array}{lcl}(\rho_1\otimes\rho_2)(x_1,x_2)(m_1\otimes m_2)&=&(\sigma_1(x_1)\otimes 1+1\otimes\sigma_2(x_2))(m_1\otimes m_2)\\&=&\sigma_1(x_1)(m_1)\otimes m_2+m_1\otimes\sigma_2(x_2)(m_2)\end{array}

where \sigma_1,\sigma_2 are the canonical inclusions of \mathfrak{g}_1 in U(\mathfrak{g}_1) and \mathfrak{g}_2 in U(\mathfrak{g}_2).  Hence M_1\otimes M_2 is both a \mathfrak{g}_1\times\mathfrak{g}_2-module and a U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)-module.

Definition 1\mathfrak{g} is solvable if its derived series D^n\mathfrak{g}=0 for some n.  It is nilpotent if its lower central series C_n\mathfrak{g}=0 for some n.  The radical of \mathfrak{g}, \sqrt{\mathfrak{g}}, is the largest solvable ideal in \mathfrak{g}.  We say \mathfrak{g} is semisimple if it has no nonzero abelian ideals.

Hence it follows that the center Z(\mathfrak{g}) of a nilpotent Lie algebra is nontrivial.  Also note that a nilpotent Lie algebra is solvable.  \mathfrak{r}=\sqrt{\mathfrak{g}} is the smallest ideal such that \mathfrak{g}/\mathfrak{r} has radical 0.

Theorem 2 (Engel).  Let V be a vector space and \mathfrak{g} be a finite dimensional subalgebra of \mathfrak{gl}(V)..  If V\neq 0, then there exists a u\neq 0 with u\in\mathfrak{g} such that xu=0 for all x\in\mathfrak{g}.

It follows that a Lie algebra \mathfrak{g} is nilpotent iff \mbox{adj}\,x is nilpotent for all x\in\mathfrak{g}.

Note that B(x,y)=\mbox{tr}(\mbox{adj}(x)\mbox{adj}(y)) defines a symmetric bilinear form on \mathfrak{g}, called the Killing form of \mathfrak{g}.  The Killing form also satisfies B(xy,z)=B(x,yz).

Proposition 3.  \mathfrak{g} is solvable iff B(D\mathfrak{g},\mathfrak{g})=0.

Proposition 4.  \mathfrak{g} is semisimple iff \mathfrak{g}=0 iff B is nondegenerate.

Theorem 5 (Weyl).  Every finite dimensional representation of a semisimple Lie algebra is completely reducible (i.e. the \mathfrak{g}-module is semisimple).

A Lie algebra is simple if its only ideals are \mathfrak{g} and 0 and noncommutative.

Proposition 6.  \mathfrak{g} is semisimple iff it is a product of simple algebras.

A Lie algebra is reductive if its adjoint representation is semisimple.  That is, it is reductive if it can be written \mathfrak{g}=\mathfrak{s}\oplus\mathfrak{a} where \mathfrak{s} is semisimple and \mathfrak{a} is abelian (since here we want \mathfrak{g} as a module to be semisimple–so that as an algebra, it has an an abelian term).  This also turns out to be equivalent to D\mathfrak{g} being semisimple.

Definition 7.  The nilpotent radical of \mathfrak{g} is the intersection of the kernels of all finite dimensional simple representations of \mathfrak{g}.

For an element x in the nilpotent radical, it follows that \mbox{adj}(x) is nilpotent.  Now for any nilpotent endomorphism u of an algebra, we have that

\displaystyle e^u=\sum_{n=0}^\infty\frac{u^n}{n!}

has a finite number of nonzero terms, and is hence well-defined.  If we can write \mathfrak{g}=\sqrt{\mathfrak{g}}\ltimes_{e^{u\cdot}}\mathfrak{s} with \mathfrak{s} semisimple and u nilpotent, then we will call \mathfrak{s} a Levi subalgebra of \mathfrak{g}.

Theorem 8 (Levi-Malcev).  Every Lie algebra has a Levi subalgebra (that is, it can be written \mathfrak{g}=\sqrt{\mathfrak{g}}\ltimes\mathfrak{s} as above).  Moreover, every other Levi subalgebra \mathfrak{s}' is the image of \mathfrak{s} under \exp(\mbox{adj}(x)) for some x in the nilpotent radical of \mathfrak{g}.

It turns out that a subalgebra of \mathfrak{g} is a Levi subalgebra iff it is a maximal semisimple subalgebra.

Theorem 9 (Ado).  Let \mathfrak{n} be the largest nilpotent ideal in \mathfrak{g}.  Then \mathfrak{g} admits a finite dimensional faithful representation \rho such that every element of \rho(\mathfrak{n}) is nilpotent.

[1]  Bourbaki, Nicholas.  Lie Groups and Lie Algebras, Chapters 1-3.  Springer-Verlag.  1971.

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