# 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)

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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