2017 Fall Workshop on Lie Algebras

Lecture 1 by D. Pham

Date: Wednesday, September 13, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • Lie algebras were motivated by relating them to Lie groups
  • The following definitions and examples were introduced:
    • Def. of Lie algebras
    • Def. of subalgebras
    • Def. of Lie algebra homomorphism
    • Def. of Lie algebra representation
    • Examples:  \(\frak{gl}\)(V), \(\frak{gl}\)(n), \(\frak{o}\)(n), \(\frak{sl}\)(n), \(\frak{sp}\)(2n)
    • Examples of Lie algebra representations: adjoint and coadjoint representations

Lecture 2 by M.S. Ravi

Date: Wednesday, October 11, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • The notion of Lie algebra ideal was introduced.  A key example is the kernel of a Lie algebra homomorphism.  Moreover, if \(\frak{g}\) is a Lie algebra and \(\frak{h}\) is an ideal of \(\frak{g}\), then \(\frak{g}/\frak{h}\) inherits a natural Lie algebra structure from \(\frak{g}\).
  • The notion of Lie algebra isomorphisms and abelian Lie algebras were defined.
  • It was noted that in dimension 1 all Lie algebras are necessarily abelian, and it was shown that in dimension 2, all non-abelian Lie algebras are isomorphic.  
  • Derived algebras were defined.  For a Lie algebra  \(\frak{g}\), its derived algebra is the ideal [\(\frak{g}\), \(\frak{g}\)].
  • Simple Lie algebras were defined.  It was shown that in dimension 3, any Lie algebra \(\frak{g}\) satisfying [\(\frak{g}\), \(\frak{g}\)]=\(\frak{g}\) is necessarily simple.

Lecture 3 by M.S. Ravi

Date: Wednesday, October 18, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • The derived series of a Lie algebra \(\frak{g}\) was defined:

\[\frak{g}:=\frak{g}^{(0)}\supset\frak{g}^{(1)}\supset \cdots \supset \frak{g}^{(n)}=\frak{g}^{(n+1)}=\cdots\]

           for some \(n\), where \(\frak{g}\)\(^{(k)}\)\(:=\)[\(\frak{g}\)\(^{(k-1)}\),\(\frak{g}\)\(^{(k-1)}\)]

  • The notion of a solvable Lie algebra was defined: a Lie algebra is solvable if \(\frak{g}\)\(^{(n)}=0\) for some \(n\).
  • Several examples of solvable Lie algebras were given:
    • Any abelian Lie algebra \(\frak{g}\) is solvable since \(\frak{g}\)\(^{(1)}=0\)
    • Any 2-dimensional non-abelian Lie algbera is solvable since \(\frak{g}\)\(^{(2)}=0\)
    • The Lie algebra \(\frak{t}\)\((n)\) of \(n\times n\) upper triangular matrices is solvable
  • Since \(\frak{g}\)':=[\(\frak{g}\),\(\frak{g}\)] is an ideal, it follows that [\(\frak{g}\),\(\frak{g}\)]=\(\frak{g}\) if \(\frak{g}\) is simple.  In particular, a simple Lie algebra cannot be solvable.
  • The normalizer and centralizer of a Lie algebra were introduced.
  • The lower central series of a Lie algebra were introduced:

\[\frak{g}:=\frak{g}^0\supset \frak{g}^1\supset \cdots \supset g^{m}=\frak{g}^{m+1}=\cdots  \]

          for some \(m\), where \(\frak{g}\)\(^k\):=[\(\frak{g}\),\(\frak{g}\)\(^{k-1}\)].

  • The notion of nilpotent Lie algebras were defined: a Lie algebra \(\frak{g}\) is nilpotent if \(\frak{g}\)\(^m=0\) for some \(m\).
  • The following inclusion was noted: \(\frak{g}\)\(^k\supset\) \(\frak{g}\)\(^{(k)}\).  Hence, ever nilpotent Lie algebra is also solvable.

Lecture 4 by M.S. Ravi

Date: Wednesday, October 18, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • For a Lie algebra \(\frak{g}\)\(^{k}\), it was noted that the \(k^{th}\) term in the derived and lower central series \(\frak{g}\)\(^{(k)}\) and \(\frak{g}\)\(^{k}\) are both ideals of \(\frak{g}\).
  • Several basic results for solvable and nilpotent Lie algebras were proved.
  • The results for solvable Lie algebras are used to prove the existence and uniqueness of Rad(\(\frak{g}\)), the radical of a Lie algebra \(\frak{g}\).  Rad(\(\frak{g}\)) is defined to be maximal solvable ideal of \(\frak{g}\).  
  • A Lie algbera is called semisimple if its radical is zero.  In particular, every simple Lie algebra is semisimple.  
  • An element \(x\) in a Lie algebra \(\frak{g}\) is called ad-nilpotent if the linear map \[ad_x: \frak{g}\longrightarrow \frak{g}\] is nilpotent.
  • Engle's Theorem: A Lie algebra \(\frak{g}\) is nilpotent iff every element of \(\frak{g}\) is ad-nilpotent.

Lecture 5 by M.S. Ravi

Date: Wednesday, October 25, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • The following theorem was proved: ``Theorem A": Let \(\frak{g}\) be a Lie subalgebra of \(\frak{gl}\)(V) which consists of nilpotent maps.  Then there exists a nonzero element \(v\in V\) such that \(xv=0\) for all \(x\in \) \(\frak{g}\).
  • The aforementioned theorem was then used to give a proof of Engel's Theorem which states any finite dimensional Lie algebra whose elements are all ad-nilpotent is itself nilpotent.
  • There was a breif discussion of Ado's Theorem which states that every finite dimensional Lie algebra is isomorphic to a Lie subalgebra of \(\frak{gl}\)(V) for some finite dimensional vector space \(V\).  In the special case when Z(\(\frak{g}\))=0, Ado's theorem is immediate since \(\frak{g}\)\(\simeq ad\)(\(\frak{g})\subset \frak{gl}(\frak{g})\).   The proof of Ado's Theorem for the case when Z(\(\frak{g}\))\(\neq 0\) is nontrivial. 

Lecture 6 by D. Pham

Date: Wednesday, November 8, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • Lie's Theorem:   Let \(\frak{g}\)\(\subset\frak{gl}\)(V) be a solvable Lie subalgebra, where V is a complex vector space.  Then there exists a non-zero \(v\in V\) and \(\lambda\in \frak{g}^\ast\) such that  \(xv=\lambda(x)v\) for all \(x\in \frak{g}\).  In other words, there exists a non-zero \(v\in V\) which is an eigenvector for every linear map \(x:V\rightarrow V\) in \(\frak{g}\); the eigenvalue of \(x\) associated to \(v\) is \(\lambda(x)\in \mathbf{C}\). 
  • The proof of Lie's theorem relies on Dynkin's Lemma: Let \(\frak{g}\)\(\subset\frak{gl}\)(V) by any Lie subalgebra.  For any ideal \(\frak{h}\) of \(\frak{g}\) and any \(\lambda\in \frak{h}^\ast\), define \[W:=\{v\in V~|~xv=\lambda(x)v,~\forall~x\in \frak{h}\}\].  Then \(W\) is \(\frak{g}\)-invariant, that is, \(xW\subset W\) for all \(x\in \frak{g}\).  Two corollaries of Lie's theorem were proved:
  • Corollary 1: Let \(\frak{g}\) be any complex solvable Lie algebra and let \[\varphi:\mathfrak{g}\rightarrow \mathfrak{gl}(V),~x\mapsto \varphi_x\] be any representation of \(\frak{g}\).  Then there exists a basis of \(V\) such that the matrix representation of \(\varphi_x\) is upper triangular.
  • Corollary 2: Let \(\frak{g}\) be a complex Lie algebra.  Then \(\frak{g}\) is solvable iff its derived algebra \(\mathfrak{g}':=[\mathfrak{g},\mathfrak{g}]\) is nilpotent.

Lecture 7 by D. Pham

Date: Wednesday, November 29, 2017

Time: 12:05pm—12:55pm

Room: S-213

Summary

  • A proof of Dynkin's lemma was given.  (This was central to the proof of Lie's theorem.)
  • The Killing form of a Lie algebra was defined.  For a Lie algebra \(\frak{g}\), the Killing form \[K: \mathfrak{g}\times \mathfrak{g}\rightarrow \mathbb{C}\] is defined by \(K(x,y):=\mbox{tr}(ad_xad_y)\).   From this definition, one sees that the Killing form is symmetric and bilinear.  Using the fact that that the adjoint map is a Lie algebra homomorphism, it also follows that \(K\) is ad-invariant: \[K([x,y],z)+K(y,[x,z])=0\].
  • The following results were stated without proof due to time constraints:
    • For any ideal \(\frak{h}\) of \(\frak{g}\), \(K_{\mathfrak{h}}=K|_{\mathfrak{h}\times \mathfrak{h}}\), where \(K_{\mathfrak{h}}\) is the Killing form of \(\frak{h}\) viewed as a Lie algebra.
    • Cartan's 1st Criterion: \(\frak{g}\) is solvable iff \(K(\mathfrak{g},\mathfrak{g}')=0\), where \(\mathfrak{g}':=[\mathfrak{g},\mathfrak{g}]\) is the derived algebra of \(\frak{g}\).
    • Cartan's 2nd Criterion:  \(\frak{g}\) is semi-simple iff \(K\) is non-degenerate.
    • \(\frak{g}\) is semi-simple iff \(\mathfrak{g}=\oplus \mathfrak{g}_i\) where each \(\mathfrak{g}_i\) is a simple ideal of \(\frak{g}\).  (This result will follow from Cartan's 2nd Criterion.)

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