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title: Moduli space of Lie algebras

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# Moduli space of Lie algebras


In this page we define the sets $V_d^\mathrm{Lie}(\mathfrak{so}(n))$ and $V_d^\mathfrak{g}(\mathfrak{so}(n))$. They depend on the integers $d,n$ and a Lie algebra $\mathfrak{g}$.

They are both subsets of the space of matrices with $\frac{n(n-1)}{2} \times d$ entries. We wonder whether they are submanifold. If so, we would like to know some information, such as their dimension and curvature.

### Special orthogonal group

Let $n > 0$ be an integer, $SO(n)$ the special orthogonal group and $\mathfrak{so}(n)$ its Lie algebra. Their dimension is $\frac{n(n-1)}{2}$.

The group $SO(n)$ is seen as the set of orthogonal $n\times n$ matrices of determinant $1$, endowed with the matrix product. From this point of view, $SO(n)$ is a compact submanifold of $M_{n,n}(\mathbb{R})$, the vector space of $n\times n$ matrices. 

Likewise, $\mathfrak{so}(n)$ is seen as the set of $n\times n$ skew symetric matrices, endowed with the bracket $[A,B] = AB - BA$. Hence $\mathfrak{so}(n)$ is a vector subspace of $M_{nn,n}(\mathbb{R})$. We also endow $\mathfrak{so}(n)$ with the Frobenius inner product $\langle A, B\rangle$.

A *Lie sub-algebra* of $\mathfrak{so}(n)$ is by definition a vector subspace $\mathfrak{h}$ of $\mathfrak{so}(n)$ stable by bracket.

### Stiefel manifold $V_d^\mathrm{Lie}(\mathfrak{so}(n))$

Let $d > 0$ be an integer.  Consider the Stiefel manifold $V_d(\mathfrak{so}(n))$, that is, the set of orthogonal $d$-frames $(A_1,\dots,A_d)$ of $\mathfrak{so}(n)$. It is a compact submanifold of the Cartesian product $\mathfrak{so}(n)^d$.

Let $m = \frac{n(n-1)}{2}$ be the dimension of $\mathfrak{so}(n)$. By fixing a basis $E_1, \dots, E_m$ of $\mathfrak{so}(n)$, we can see $V_d(\mathfrak{so}(n))$ as a subset of $M_{m,d}(\mathbb{R})$, the vector space of $m\times d$ matrices.

A $d$-frame $(A_1,\dots,A_d) \in V_d(\mathfrak{so}(n))$ defines a $d$-dimensional vector subspace of $\mathfrak{so}(n)$, but it may not be stable by bracket. Hence we also define $V_d^\mathrm{Lie}(\mathfrak{so}(n))$, the set of orthogonal $d$-frames whose spanned vector subspace is a Lie algebra.

**Scholie:** $V_d^\mathrm{Lie}(\mathfrak{so}(n)) \subset M_{m,d}(\mathbb{R})$ is the zero set of a collection of polynomials.
To see so, consider a $m\times d$ matrix $A$. We write
$$
A 
= (A_1,\dots,A_d)
= \begin{pmatrix} 
    a_1^1 & \dots  & a_{d}^1\\
    \vdots & \ddots & \vdots\\
    a_{1}^m & \dots  & a_{d}^m 
    \end{pmatrix}.
$$
The matrix $A$ represent a $d$-frame if and only if 
$$
A^\mathrm{T} A = I_d.
$$
Besides, the product $P_A = A (A^\mathrm{T}A)^{-1}A^\mathrm{T}$ is a $m \times m$ matrix, representing the orthogonal projection on the subspace spanned by $(A_1,\dots,A_d)$. Now, the subspace spanned by $A$ is stable by bracket if and only if
$$
P_A [A_i, A_j] = [A_i, A_j] 
$$
for all $1 \leq i < j \leq d$. 
In other words, $V_d^\mathrm{Lie}(\mathfrak{so}(n))$ is the subspace of $M_{m,d}(\mathbb{R})$ consisting of elements $A = (a_i^j)$ which are common roots of the polynomials
- $A^\mathrm{T} A - I_d$,
- $P_A [A_i, A_j] - [A_i, A_j]$ for all $1 \leq i < j \leq d$.

The first polynomial has degree $2$, and the second polynomials have degree $8 = 4 \times 2$.

**Question:** Is $V_d^\mathrm{Lie}(\mathfrak{so}(n))$ a submanifold of $M_{m,d}(\mathbb{R})$?

### Stiefel manifold $V_d^\mathfrak{g}(\mathfrak{so}(n))$

Let $\mathfrak{g}$ be any Lie algebra of dimension $d$, and $(B_1, \dots, B_n)$ a basis. Let $(c_{i,j}^k)$ be its structure constants, defined as
$$
[B_i, B_j] = \sum_{k = 1}^d c_{i,j}^k B_k
$$
for all $1 \leq i < j \leq d$.

We define $V_d^\mathfrak{g}(\mathfrak{so}(n))$ as the subspace of  $V_d^\mathrm{Lie}(\mathfrak{so}(n))$ consisting of $d$-frames that admit the same structure constants. That is, $A = (A_1,\dots,A_d)$ belongs to $V_d^\mathfrak{g}(\mathfrak{so}(n))$ if 
$$
[A_i, A_j] = \sum_{k = 1}^d c_{i,j}^k A_k
$$
for all $1 \leq i < j \leq d$.

**Scholie:** $V_d^\mathfrak{g}(\mathfrak{so}(n)) \subset M_{m,d}(\mathbb{R})$ is the zero set of a collection of polynomials.
Indeed, $V_d^\mathfrak{g}(\mathfrak{so}(n))$ is the subspace of $M_{m,d}(\mathbb{R})$ consisting of elements $A = (a_i^j)$ which are common roots of the polynomials
- $A^\mathrm{T} A - I_d$,
- $[A_i, A_j] - \sum_{k = 1}^d c_{i,j}^k A_k$ for all $1 \leq i < j \leq d$.

The first polynomial has degree $2$, and the second polynomials have degree $2$.

**Question:** Is $V_d^\mathfrak{g}(\mathfrak{so}(n))$ a submanifold of $M_{m,d}(\mathbb{R})$?