1
$\begingroup$

I wanted to ask about something confusing in my Lie Algebras notes.

At one point we define the Lie Algebra $\mathfrak{gl}(V)$ to be the vector space $GL_n(V)$ with the bracket $[x , y ] = xy - yx$. (Here $V$ is an $n$-dimensional vector space over a field, $k$.)

Later, it says "let $\mathfrak{gl}(V) = End(V)$ with bracket given by $[x,y] = xy - yx$.

The use of the word "let" here confuses me. Given the same notation of $\mathfrak{gl}(V)$ I expect that these two Lie Algebras should be equivalent.

However, my understanding is that $GL_n(V)$ is the vector space of invertible $n \times n$ matrices over the field $k$, with respect to a chosen basis of $V$, and that $End(V)$ is simple the vector space of Endomorphisms of $V$.

However, this definition of $\mathfrak{gl}(V)$ seems to imply that, as vector spaces, $End(V) = GL_n(V)$, but this then implies that everything in $End(V)$ is invertible, which is not the case.

I am very confused by this. Should the $End(V)$ be replaced with $Aut(V)$? Am I misunderstanding something simple about Lie Algebras, $GL_n(V)$ or $End(V)$?

I would really appreciate some help understanding this, thank you.

$\endgroup$
  • 5
    $\begingroup$ Is $GL_n(V)$ a vector space? $\endgroup$ – Hugo Oct 10 '18 at 9:50
4
$\begingroup$

The first reference to $GL$ is wrong and the reference to $End$ is correct. Indeed, $GL(V)$ is not preserved by the bracket since for instance a matrix bracketed with itself is not invertible.

$\endgroup$
3
$\begingroup$

There is a misteka from the start: $\mathfrak{gl}(V)$ is the vector space of all endomorphisms of $V$. In fact, $GL_n(V)$ isn't even a vector space.

And the bracket operation on $\operatorname{End}(V)$ should read: $[x,y]=x\circ y-y\circ x$.

$\endgroup$
  • $\begingroup$ Thank you for the answer and clarification on the bracket. The bracket also confused me because I wasn't sure what composition or multiplication meant in the vector space $\mathfrak{gl}(V)$. Is it correct to say that $\mathfrak{gl}(V)$ satisfies all the conditions of being a vector space, but ALSO has another operation other than addition defined for it? $\endgroup$ – user366818 Oct 10 '18 at 9:59
  • 1
    $\begingroup$ @user366818 Indeed. Besides addition, $\mathfrak{gl}(V)$ also has the composition operation. $\endgroup$ – José Carlos Santos Oct 10 '18 at 10:00
2
$\begingroup$

Complementary info, where the confusion may come from: uppercase letters are used to denote Lie groups, while lowercase letters are used to denote their associated Lie algebras. $GL_n(V)$ (not a vector space) is the Lie group of invertible $n\times n$ matrices from End$(V)$, while $gl_n(V)$ is its associated Lie algebra (the vector space End$(V)$ along with the commutator). If instead you pick the Lie group $SL_n(V)$ of invertible matrices with determinant 1, then the associated Lie algebra $sl_n(V)$ is composed of those matrices with trace $0$ (and the commutator as product).

$\endgroup$
  • 2
    $\begingroup$ Thank you, this makes it clear that although $\mathfrak{gl}(V)$ is the associated Lie Algebra of $GL_n(V)$, they do not have the same elements, which is an assumption I had made. $\endgroup$ – user366818 Oct 10 '18 at 10:02
  • 2
    $\begingroup$ @user366818 The usual interpretation is that the Lie algebra is the tangent space of the (connected component of the identity element of the) Lie group. So, although a group is not a vector space, its Lie algebra indeed is, with vectors pointing to local directions inside the group (but enriched with a new interesting product for vectors, the commutator). I tend to visualize a sphere with a plane on top... $\endgroup$ – Jose Brox Oct 10 '18 at 11:50
  • $\begingroup$ Thank you for the explanation. In the course I am taking I believe we are trying to define everything strictly algebraically, and so I believe we are avoiding this interpretation that the Lie algebra is the tangent space of the Lie group. $\endgroup$ – user366818 Oct 10 '18 at 11:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.