Cartan subalgebras and root systems

A Cartan matrix $A$ is a square matrix whose elements $a_{ij}$ satisfy the following conditions:

1. $a_{ij}$ is an integer, one of $\left\{ -3,-2,-1,0,2\right\}$

2. $a_{jj} = 2$ for all diagonal elements of $A$

3. $a_{ij} \leq 0$ off of the diagonal

4. $a_{ij} = 0$ iff $a_{ji} = 0$

5. There exists an invertible diagonal matrix $D$ such that $DAD^{-1}$ gives a symmetric and positive definite quadratic form. Give a $2 \times 2$ non-diagonal Cartan matrix and its eigenvalues.

The roots of a semisimple Lie algebra are the Lie algebra weights occurring in its adjoint representation. The set of roots forms the root system, and is completely determined by the semisimple Lie algebra. Consider the semisimple Lie algebra $\mathfrak {sl}(2,\mathbb {R})$ with the generators

Find the roots.

A Chevalley basis for the semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {R})$ is given by

where $Y_{j} = X_{j}^{T}$ for $j = 1,2,3$. The Lie algebra has rank 2 owing to $H_{1}$, $H_{2}$ and $[H_{1},H_{2}] = 0_{3}$. Another basis could be formed by looking at the linear combinations

Consider the semisimple Lie algebra $\mathfrak {sl}(n+1,\mathbb {F})$. Let $E_{i,j}$ ($i,j \in \{ 1,2,\ldots ,n+1\}$) denote the standard basis, i.e. $(n+1) \times (n+1)$ matrices with all entries zero except for the entry in the $i$-th row and $j$-th column which is one. We can form a Cartan-Weyl basis with

Show that $E_{i,j}$ are root vectors for $i \neq j$, i.e. there exists $\lambda_{H,i,j} \in \mathbb {F}$ such that

for all $H \in \operatorname {span}\{ \, H_{1}, \ldots , H_{n} \, \}$.

The Cartan matrix $A$ of a rank $r$ root system of a semi-simple Lie algebra is an $r \times r$ matrix whose entries are derived from the simple roots. The entries of the Cartan matrix are given by

where $(\, , \, )$ is the Euclidean inner product and $\alpha_{j}$ are the simple roots. The entries are independent of the choice of simple roots (up to ordering).

The semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {R})$ has dimension 8. The standard basis is given by

Find the commutator table. Find the rank of the Lie algebra.

There is only one simple complex Lie algebra of rank 1. If $\alpha = \lambda e_{1}$ is a nonzero root, $-\lambda e_{1}$ is the other. The canonical choice for $\lambda$ is $1/\sqrt{2}$. The commutators for the basis

are

Find a $2 \times 2$ matrix representation for this semisimple Lie algebra.

Consider the vectors in $\mathbb {R}^{2}$

Then

Find eight $3 \times 3$ matrices

such that the following commutation relations are satisfied

and $[H_{1},H_{2}] = 0$. We can assume that $H_{1}, H_{2}$ are diagonal matrices.

Show that the dimension of a Cartan sub-Lie algebra is not in general the maximal dimension of an abelian sub-Lie algebra. Consider the Lie algebra $\mathfrak {sl}(2n,\mathbb {C})$ of $2n \times 2n$ matrices of trace 0.

The semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {C})$ has dimension 8 and rank 2. The standard basis is given by

Note that $e_{1} = f_{1}^{T}$, $e_{2} = f_{2}^{T}$, $e_{13} = f_{13}^{T}$. Find the commutation relation table.

Consider the Lie algebra $\mathfrak {sl}(3,\mathbb {C})$. Given the roots find the structure constants. The semisimple Lie algebra $\mathfrak {sl}(3,\mathbb {C})$ has dimension 8 and rank 2. The root system of $\mathfrak {sl}(3,\mathbb {C})$ consists of 6 vectors which go from the origin to vertices of regular hexagon. Check the Jacobi identities for the resulting structure constants.

Consider the semisimple Lie algebra $\mathfrak {sl}(n+1,\mathbb {F})$. Let $E_{i,j}$ ($i,j \in \{ 1,2,\ldots ,n+1\}$) denote the standard basis, i.e. $(n+1) \times (n+1)$ matrices with all entries zero except for the entry in the $i$th row and $j$th column which is one. We can form a Cartan-Weyl basis with

Show that $E_{i,j}$ are root vectors for $i \neq j$, i.e. there exists $\lambda_{H,i,j} \in \mathbb {F}$ such that

for all $H \in \left\langle H_{1},\ldots ,H_{n}\right\rangle$.

The exceptional Lie algebra $\mathfrak {g}_{2}$ has the simple roots

Find the Cartan matrix. Verify the properties of the Cartan matrix.

The exceptional Lie algebra $\mathfrak {g}_2$ has rank 2 and dimension 14. A basis is given by

with the commutation relations $[H_1,H_2] = 0$ and

Thus

Thus $H_1, X_1, Y_1$ and $H_2, X_2, Y_2$ each span the sub Lie algebra $\mathfrak {sl}(2,\mathbb {C})$ of $\mathfrak {g}_2$. The commutator table for $\mathfrak {g}_2$ (Table 4.1) is, where entry $(i,j)$ gives $[\text{row}_i, \text{column}_j]$ for $i < j$ and is left blank (not part of the table) otherwise:

Note that $\mathfrak {sl}(3,\mathbb {C})$ is a sub Lie algebra of $\mathfrak {g}_2$. Write a SymbolicC++ program that finds the adjoint representation.