Lie Algebras

Now we're going to talk about a subject closely related to matrix groups using the infinitesimals that we established.

So let's go back to our idea of matrix groups. We are considering \(n \times n\) matrices that are invertible, and we're in particular considering subsets of such matrices that obey the various group laws. We're going to say that our matrices have entries in either \(\mathbb{R}\) or \(\mathbb{C}\), and when I don't want to specify one or the other I'll say that the entries are in \(\mathbb{k}\).
Recall that every group has an identity element that we'll call \(I\), which here means the \(n \times n\) matrix with \(1\)s down the diagonal and \(0\)s everywhere else. We want to look at group elements of the form \(I + sX\), where \(s\) is an infinitesimal and \(X\) has either real or complex entries. This is the first instance of "things that don't technically exist" regarding infinitesimals; if our group is made of matrices only with real or complex entries, then we don't actually have elements of the form \(I + sX\) since that would require infinitesimal entries.
But we're going to pretend as if \(I + sX\) is in the group, the same way that we're pretending that our numbers allow for infinitesimals at all. There are ways to do this more formally, via what are called "algebraic schemes" but I'm not going to get into that.
To add in a condition that will make this pretending a little less egregious, we're going to say that if \(I + sX\) is in the group, then \(I + csX\) is in the group for any number \(c \in \mathbb{k}\).
Note that we already know that \(I + nsX\) is in the group for any integer \(n\): if we multiply out \((I + sX)^n\) for positive integer \(n\) we get \(I + nsX\), and the inverse of \(I + sX\) is \(I - sX\) (Check this yourself!), so \(I + nsX\) is in the group for any negative integer \(n\). But we extend to any number in \(k\) to get ourselves a Lie group.
We want to look at the set of matrices \(X\) with entries in \(\mathbb{k}\) such that \(I\) plus an infinitesimal times \(X\) is in \(G\). We call this set \(\mathfrak{g}\). What is in this set? We know that if \(X\) is in \(\mathfrak{g}\) then \(cX\) is for any \(c\in \mathbb{k}\). If we look at \(Y\) such that \(I + sY\) is in the group, then we get that
$$(I + sX)(I + sY) = I + s(X + Y)$$ so \(X + Y\) is in \(\mathfrak{g}\). Hence \(\mathfrak{g}\) is a vector space.
\(\mathfrak{g}\) has some further structure, though.
Consider the question of whether \(G\) is commutative or not, i.e. whether for \(g\) and \(h\) in \(G\), \(gh = hg\). We can look at this question as instead asking whether \(ghg^{-1}h^{-1} = I\), since if we can swap \(g\) and \(h\), then \(ghg^{-1}h^{-1} = hgg^{-1}h^{-1} = hh^{-1} = I\), and conversely.
Now let's look at group elements of the form \(I + sX\). Recall that \((I + sX)^{-1} = I - sX\). We get
$$(I + sX)(I + sY)(I - sX)(I - sY) = I$$ so that doesn't tell us anything useful. But remember that we have two independent infinitesimals to play around with, so we look instead at \(I + tY\) and get
$$(I + sX)(I + tY)(I - sX)(I - tY) = I + st(XY - YX)$$ So now the object \(st(XY - YX)\) measures whether \(I + sX\) and \(I + tY\) commute or not.
Now \(st\) is an infinitesimal, and \(I + st(XY - YX)\) is in the group, so we get that \(XY - YX\) is in \(\mathfrak{g}\). This is the extra structure that \(\mathfrak{g}\) has, called a Lie bracket, denoted by \([X, Y] = XY - YX\).
Abstracting away from our groups and matrices, we get the abstract rules for a Lie algebra:
1): A Lie algebra \(\mathfrak{g}\) is a vector space with an operation denoted by \([, ]\).
2): If \(X\) and \(Y\) are in \(\mathfrak{g}\) then \([X, Y]\) is in \(\mathfrak{g}\).
3): Bilinearity: For \(X, Y,\) and \(Z\) in \(\mathfrak{g}\) and \(c \in \mathbb{k}\),
$$[X, Y + cZ] = [X, Y] + c[X, Z], [X + cY, Z] = [X, Z] + c[Y, Z]$$ 4): Antisymmetry: For \(X\) and \(Y\) in \(\mathfrak{g}\), \([X, Y] = -[Y, X]\)
5): Jacobi identity: For \(X, Y\) and \(Z\) in \(\mathfrak{g}\),
$$[X, [Y, Z]] = [[X, Y], Z] + [Y, [X, Z]]$$ You can check that all of these hold for matrices, but sometimes we want to talk about Lie algebras without going through matrices, just as we have groups that are sets without being realized as matrices.
Let's look at some examples. Recall the group \(SO_3(\mathbb{R})\) of rotations in 3-dimensional real space. The elements of \(SO_3(\mathbb{R})\) obey the rule \(T^t T = I\) where \(^t\) indicates matrix transposition.
If we replace \(T\) with \(I + sX\), we get the rule \((I + sX)^t (I + sX) = I\). \((I + sX)^t = I + sX^t\), so we get
$$(I + sX^t)(I + sX) = I$$ We already know that \((I - sX)(I + sX) = I\), so we get the rule that \(X^t = -X\). So the Lie algebra of \(SO_3(\mathbb{R})\), often denoted by \(so_3(\mathbb{R})\), is given by the \(3 \times 3\) antisymmetric matrices with real entries.
Another example: the group \(SL_n\) of \(n \times n\) matrices with determinant \(1\). Given an element \(I + sX\), what is the determinant?
The determinant of \(I + sX\) ends up being \(1 + s\text{tr}(X)\), which you can see by writing out a few of the terms and noting that any term that involves \(X\) more than once has to vanish since \(s^2 = 0\). So if \(I + sX\) has determinant \(1\), that implies that \(tr(X)\) is \(0\), and conversely. So the Lie algebra \(sl_n\) is the set of \(n \times n\) matrices with trace \(0\).
For \(GL_n\), \(X\) can be anything, since \(I + sX\) always has inverse \(I - sX\) for any \(X\).
This, by the way, is how we're going to get around the issues of whether these infinitesimal elements are actually in the group or not. We look at matrices with entries not in \(\mathbb{k}\), but rather in \(\mathbb{k}[s, t]\) so that we have infinitesimals, and then we focus on groups of the form "matrices obeying these conditions" where the conditions can be written as polynomials in the entries of the matrices. In this setup, we actually do end up with elements of the form \(I + sX\).
If we wanted to do this without infinitesimals, we'd have to talk about derivatives and one-parameter groups and invariant vector fields and all that machinery from analytic differential geometry, and honestly the synthetic, infinitesimal stuff doesn't get enough love.