lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: ec2335c580aa04c0d9f793327cef4b8b59fcb97b
Parent: 5e6751ba4bc2f67684c0e19ec2a6bf0be878857e
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 27 Nov 2022 00:59:35 +0000

Fixed multiple typos

Diffstat

3 files changed, 5 insertions, 5 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	4	2	2
Modified	sections/semisimple-algebras.tex	2	1	1
Modified	sections/sl2-sl3.tex	4	2	2

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -18,7 +18,7 @@ canonical basis elements \(e_1, \ldots, e_n \in K^n\). In particular, a
 1-dimensional representation of \(K^n\) is just a choice of \(n\) scalars
 \(\lambda_1, \ldots, \lambda_n\). Different choices of scalars yield
 non-isomorphic representations, so that the 1-dimensional representations of
-\(K^n\) are parametrized by points in \(K^n\).
+\(K^n\) are parameterized by points in \(K^n\).
 
 This goes to show that classifying the representations of Abelian algebras is
 not that interesting of a problem. Instead, we focus on the the
@@ -360,7 +360,7 @@ compact form, whose representations are known to be completely reducible -- see
 
 This proof, however, is heavily reliant on the geometric structure of
 \(\mathbb{C}\). In other words, there is no hope for generalizing this for some
-arbitrary \(K\). Furnately for us, there is a much simpler, completely
+arbitrary \(K\). Fortunately for us, there is a much simpler, completely
 algebraic proof of complete reducibility, which works for algebras over any
 algebraically closed field of characteristic zero. The algebraic proof included
 in here is mainly based on that of \cite[ch. 6]{kirillov}, and uses some basic

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -12,7 +12,7 @@ This was simple enough to do in the case of \(\mathfrak{sl}_2(K)\), but the
 reasoning behind it, as well as the mere fact equation (\ref{sym-diag}) holds,
 are harder to explain in the case of \(\mathfrak{sl}_3(K)\). The eigenspace
 decomposition associated with an operator \(V \to V\) is a very well-known
-tool, and readers familizared with basic concepts of linear algebra should be
+tool, and readers familiarized with basic concepts of linear algebra should be
 used to this type of argument. On the other hand, the eigenspace decomposition
 of \(V\) with respect to the action of an arbitrary subalgebra \(\mathfrak{h}
 \subset \mathfrak{gl}(V)\) is neither well-known nor does it hold in general:

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -203,12 +203,12 @@ computing
 \]
 we can see that \((V \oplus W)_\lambda = V_\lambda + W_\lambda\). Hence the set
 of eigenvalues of \(h\) in a representation \(V\) is the union of the sets of
-eigenvalues in its irreducible components, and the correspoding eigenspaces are
+eigenvalues in its irreducible components, and the corresponding eigenspaces are
 the direct sums of the eigenspaces of such irreducible components.
 
 In particular, if the eigenvalues of \(V\) all have the same parity -- i.e.
 they are either all even integers or all odd integers -- and the dimension of
-each eigenspace is no greather than \(1\) then \(V\) must be irreducible, for
+each eigenspace is no greater than \(1\) then \(V\) must be irreducible, for
 if \(U, W \subset V\) are subrepresentations with \(V = W \oplus U\) then
 either \(W_\lambda = 0\) for all \(\lambda\) or \(U_\lambda = 0\) for all
 \(\lambda\).