Lie Algebras Constructed From Graphs, by A.J. Butler

Definition: Graph Isomorphism

Graphs $G$ and $G'$ are isomorphic if we can construct a bijection between vertex sets $f: V\to V'$ s.t. $u, v \in V$ are adjacent iff $f(u), f(v) \in V'$ are adjacent
i.e. adjacent vertices are mapped to adjacent vertices

Definition: Lie Algebra

A Lie Algebra $\frak{g}$ is a vector space with a bilinear operation $[, ]: \frak{g}\times \frak{g} \to \frak{g}$ called the Lie bracket with
1) $[x, x] = 0$ for all $x \in \frak{g}$
2) $[x, y] = -[y, x]$ (implied by 1 and bilinearity)
3) $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ for all $x, y, z \in g$

Example: We see that $\mathbb{R}^3$ with cross product is a Lie algebra
$A \times B = -B \times A$ according to right hand rule, and we can validate that the Jacobi identity works

Another example: If $A$ is an associative $\mathbb{F}$-algebra (i.e. an $\mathbb{F}$-vector space equipped with bilinear associative operation $\ast$).
For all $v, w\in A$, define the commutator $[v, w] \coloneqq v\ast w - w\ast v$.
An explicit example of this would be $\frak{gl}_n(\mathbb{R})$, the set of all $n\times n$ matrices over $\mathbb{R}$ with the Lie brackets defined as $[A, B] = A\ast B - B \ast A$.

Definition: Just Some Notations/ideals

• Let $\frak{g} = \frak{g}^0 = \frak{g}_0$
• Let $\frak{g}^{n+1} = [\frak{g}, \frak{g}^n]$
• Let $\frak{g}_{n+1} = [\frak{g}_n, \frak{g}_n]$
  We construct the following sequences:
  

  $$\begin{align*} \frak{g}^0 \supseteq \frak{g}^1 \supseteq \frak{g}^2 \supseteq \dots \text{ (lower central series)} \\ \frak{g}_0 \supseteq \frak{g}_1 \supseteq \frak{g}_2 \supseteq \dots \text{ (derived series)} \\ \end{align*}$$

  
  A Lie algebra is nilpotent if $\frak{g}^n = 0$ for some integer $n$.
  A Lie algebra is solvable if $\frak{g}_n = 0$ for some integer $n$.

Definition: Lie Algebra Isomorphisms

A **Lie Algebra Isomorphism is a Lie Algebra Homomorphism that is also a Vector Space Isomorphism.
Formally, $\phi: \frak{g}\to \frak{h}$ s.t. $[\phi(x), \phi(y)] = \phi([x, y])$ for all $x, y \in \frak{g}$
Vector space isomorphism: Linear, bijection, thus dim equal and $\phi(v) = 0$ implies $v = 0$

Examples:
- Identity map $x \mapsto x$ is a Lie Algebra to itself
- If the brackets are trivial ($[v_1, v_2] = 0 \forall v_1, v_2 \in V$ and $[w_1, w_2]=0\forall w_1, w_2\in W$), then they are isomorphic iff dim are the same

Definition: Lie Algebra Associated On a Graph

Let $G = (V, E)$ be a simple graph (no self loops). Let $\frak{g} = \text{span}_{\mathbb{R}}(V\bigcup E)$ be a vector space.
We define the bracket by

$$[v_i, v_j] = \begin{cases} e_k &\text{if } i < j \text{ and } e_k \text{ is an edge between } v_i \text{ and } v_j \\ -e_k &\text{if } i > j \text{ and } e_k \text{ is an edge between } v_i \text{ and } v_j \\ 0 & \text{otherwise}\\ \end{cases}$$

.#-> Related to cohomology? Holden: Yes, but not in the way you think it is
.#-> A.J.: The graph structure informs the cohomology of the Lie Algebra of the graph

All other brackets not defined by linearity are zero.
The vector space $\frak{g}$ with this braket operation is the Lie algebra associated with $G$.
Because it acts as a vector space, we can have linear combinations and it's more powerful than just an adjacency matrix