# Finite-dimensional Semisimple Lie Algebras over $\CC$ (Wednesday, August 17)

## Humphreys 1.1

:::{.remark}
Main goal: understand semisimple finite-dimensional Lie algebras over $\CC$.
These are extremely well-understood, but there are open problems in infinite-dimensional representations, representations over other fields, and Lie superalgebras.
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:::{.remark}
Recall that an associative algebras is a ring with the structure of a $k\dash$vector space, and *algebra* generally means a non-associative algebra.
Given any algebra, one can define a new bilinear product
\[
[\wait, \wait]: A\tensor_k A &\to A \\
a\tensor b &\mapsto ab-ba
\]
called the **commutator bracket**.
This yields a new algebra $A_L$ which is an example of a Lie algebra.
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:::{.definition title="Lie algebra"}
For $L\in \mods{\FF}$ with an operation $[\wait, \wait]: L\times L\to L$ (called the **bracket**) is a **Lie algebra** if

1. $[\wait,\wait]$ is bilinear,
2. $[x, x] = 0$ for all $x\in L$, and
3. the Jacobi identity holds: $[x[yz]] + [y[zx]] + [z[xy]] = 0$.

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:::{.exercise title="?"}
Check that $[ab]\da ab-ba$ satisfies the Jacobi identity.
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:::{.remark}
\envlist

- Expanding $[x+y, x+y] = 0$ yields $[xy] = -[yx]$.
  Note that this is equivalent to axiom 2 when $\characteristic \FF\neq 2$ (given axiom 1).

- The Jacobi identity can be rewritten as $[x[yz]] = [[xy]z] + [y[xz]]$, where the second term is an error term measuring the failure of associativity.
  Note that this is essentially the Leibniz rule.

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:::{.definition title="Abelian Lie algebras"}
A Lie algebra $L\in\Lie\Alg\slice\FF$ is **abelian** if $[xy]=0$ for all $x,y\in L$.
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:::{.definition title="Morphisms of Lie algebras"}
A **morphism** in $\Lie\Alg\slice\FF$ is a morphism $\phi\in \mods{\FF}(L, L')$ satisfying $\phi( [xy] ) = [ \phi(x) \phi(y) ]$.
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:::{.exercise title="?"}
Check that if $\phi$ has an inverse in $\mods{\FF}$, then $\phi$ automatically has an inverse in $\Lie\Alg\slice \FF$.
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:::{.definition title="Subobjects"}
A vector subspace $K\leq L$ is a **Lie subalgebra** if $[xy]\in K$ for all $x,y\in K$.
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:::{.remark}
\envlist

- Note that any nonzero $x\in L$ determines a 1-dimensional Lie subalgebra $K\da \FF\cdot x$, which is in fact abelian.
- A big source of Lie algebras: left-invariant vector fields on a Lie group.
- We'll restrict to finite-dimensional algebras for the remainder of the class.

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## Humphreys 1.2: Linear Lie Algebras

:::{.remark}
For $V\in \mods{\FF}$, the endomorphisms $A\da \Endo_\FF(V)$ is an associative algebra over $\FF$.
Thus it can be made into a Lie algebra $\liegl(V) \da A_L$ by defining $[xy] = xy-yx$ as above.
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:::{.definition title="Linear Lie algebras"}
Any subalgebra $K\leq \liegl(V)$ is a **linear Lie algebra**.
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:::{.remark}
After picking a basis for $V$, there is a noncanonical isomorphism $\Endo_\FF(V) \cong \Mat_{n\times n}(\FF)$ where $n\da \dim_\FF V$.
The resulting Lie algebra is $\liegl_n(\FF) \da \Mat_{n\times n}(\FF)_L$.
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:::{.fact}
By Ado-Iwasawa, any finite-dimensional Lie algebra is isomorphic to some linear Lie algebra.
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:::{.example title="?"}
The upper triangular matrices form a subalgebra $\liet_n(\FF) \leq \liegl_n(\FF)$.[^commutator_triangle]
This is sometimes called the Borel and denoted $\lieb$.
There is also a subalgebra $\lien_n(\FF)$ of strictly upper triangular matrices.
The diagonal matrices form a maximal torus/Cartan subalgebra $\lieh_n(\FF)$ which is abelian.

[^commutator_triangle]: 
You get something interesting if you take the commutator bracket of two upper triangular matrices.

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:::{.example title="Classical Lie algebras"}
\envlist

- Type $A_n \leadsto \liesl_{n+1}(\FF)$ is the special linear Lie algebra, traceless matrices.
- Type $B_n \leadsto \lieso_{2n+1}(\FF)$ is the odd orthogonal Lie algebra.
- Type $C_n \leadsto \liesp_{2n}(\FF)$ is the symplectic Lie algebra.
- Type $D_n \leadsto \lieso_{2n}(\FF)$ is the even orthogonal Lie algebra.
- The remaining 3 are defined by matrices satisfying $sx = -x^t s$ where $s$ is one of the following:
  - $\mattt 1 0 0 0 0 {I_n} 0 {I_n} 0$ corresponding to $\lieso_{2n+1}$,
  - $\matt 0 {I_n} {-I_n} 0$ corresponding to $\liesp_{2n}$,
  - $\matt 0 {I_n} {I_n} 0$ corresponding to $\lieso_{2n}$.

These can be viewed as the matrices of a nodegenerate bilinear form: writing $N$ for the size of the matrices, the matrices act on $V \da \FF^N$ by a bilinear form $f: V\times V\to \FF$ given by $f(v, w) = v^t s w$.
The form will be symmetric for $\lieso$ and skew-symmetric for $\liesp$.
The equation $sx=-x^ts$ is a version of preserving the bilinear form $s$.
Note that these are the Lie algebras of the Lie groups $G = \SO_{2n+1}(\FF), \Symp_{2n}(\FF), \SO_{2n}(\FF)$ defined by the condition $f(gv, gw) = f(v, w)$ for all $v,w\in \FF^N$ where $G = \ts{g\in \GL_N(\FF) \st f(gv, gw) = f(v, w)}$.
This is equivalent to the condition that $f(gv, w) = f(v, g\inv w)$.
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:::{.remark}
Philosophy: $G\to \lieg$ sends products to sums.
\todo{Check, might have gotten this backward.}
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:::{.exercise title="?"}
Check that the definitions of $\SO_n(\FF), \Symp_n(\FF)$ yield Lie algebras.
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