Problem 36 - Algebra

Question

Let \(F\) be a field of characteristic \(0 .\) Let \(g=5 l_{n}(F)\) be the vector space of matrices with trace 0 , with its Lie algebra structure \([X, Y]=X Y-Y X\). Let \(E_{i j}\) be the matrix having \((i, j)\) -component 1 and all other components \(0 .\) Let \(G=S L_{n}(F) .\) Let \(A\) be the multiplicative group of diagonal matrices over \(F\). (a) Let \(H_{i}=E_{i i}-E_{i+1, i+1}\) for \(i=1, \ldots, n-1\). Show that the elements \(E_{j}\) \((i \neq j), H_{1}, \ldots, H_{n-1}\) form a basis of \(g\) over \(F .\) (b) For \(g \in G\) let \(\mathbf{c}(g)\) be the conjugation action on \(g\). that is \(\mathbf{c}(g) X=g X g^{-1}\). Show that each \(E_{j}\) is an cigenvector for this action restricted to the group \(A\). (c) Show that the conjugation representation of \(G\) on \(g\) is irreducible, that is, if \(V \neq 0\) is a subspace of \(\mathrm{g}\) which is \(\mathrm{c}(G)\) -stable, then \(V=\mathrm{g} .\) Hint: Look up the sketch of the proof in [JoL 01], Chapter VII, Theorem \(1.5\), and put in all the details. Note that for \(i \neq j\) the matrix \(E_{y}\) is n?lpotent, so for variable \(t\). the exponential series \(\exp \left(t E_{y}\right)\) is actually a polynomial. The derivative with respect to \(t\) can be taken in the formal power series \(F[[t]]\), not using limits. If \(X\) is a matrix, and \(x(t)=\exp (t X)\), show that $$ \left.\left.\frac{d}{d t} x(t) Y x(t)^{-1}\right|_{t=0}=X Y-Y X=\mid X, Y\right] $$

Step-by-Step Solution

Answer

In summary, we have shown the following: (a) Elements $E_{ij}$ $(i \neq j)$, $H_1, ..., H_{n-1}$ form a basis of the Lie algebra $g$ over $F$ by proving that they are linearly independent and span the space of matrices with trace 0. (b) Each $E_{ij}$ (with $i \neq j$) is an eigenvector for the conjugation action restricted to the group $A$, as demonstrated by computing the conjugation action on the given matrices. (c) The conjugation representation of $G$ on $g$ is irreducible. We used the derivative of the conjugation action with respect to $t$ and the Lie bracket to show that if $V$ is a non-zero subspace of $g$ that is stable under the conjugation action, then $V = g$.

Step 1: Recall the properties of the trace.

From linear algebra, we know that the trace of a matrix is the sum of its diagonal elements. Additionally, two important properties of the trace are: 1. Tr(A + B) = Tr(A) + Tr(B) 2. Tr(A * B) = Tr(B * A) Now, we want to show that the elements \(E_{ij}\) \( (i \neq j) \), \(H_1, ..., H_{n-1}\) form a basis of the Lie algebra g over F.

Step 2: Confirming they form a basis

To do that, we'll check two properties: 1. They are linearly independent. 2. They span the space of matrices with trace 0. Given that the vector space of matrices with trace 0 is of dimension \(n^2 - n\), the elements \(E_{ij}\) when \(i \neq j\) provide \((n^2 - n^2)+n = n\) basis elements. Since all \(E_{ij}\) have trace 0 and are clearly linearly independent (each having a single non-zero entry at different positions), that part of the basis is established. Now, for the \(H_i = E_{ii} - E_{i+1, i+1}\) with \(i = 1, \ldots, n-1\), we have \(n - 1\) elements, each with trace 0. These elements are also linearly independent because each \(H_i\) has non-zero values in positions \((i, i)\) and \((i+1, i+1)\), with the rest being 0. The total number of elements in the basis is \(n + (n - 1) = n^2 - 1\), confirming that the given set of elements form a basis of the Lie algebra g when the field F has characteristic 0. (b) Eigenvectors for the conjugation action

Step 1: Define the conjugation action

The conjugation action is defined as \(\textbf{c}(g)X = gXg^{-1}\), where g belongs to the group G.

Step 2: Compute the conjugation action for each \(E_{ij}\)

For \(A \in A\), we have \(A = diag(a_1, a_2, \ldots, a_n)\) with \(a_i \neq 0\) for all i. We want to show that each \(E_{ij}\) (with \(i \neq j\)) is an eigenvector for the conjugation action restricted to A. To do this, compute the conjugation action: \[ \textbf{c}(A)E_{ij} = AE_{ij}A^{-1} \] Since A is a diagonal matrix, we just need to consider the non-zero entries of E_{ij}. The result will be a matrix with a non-zero entry at position \((i,j)\) and a value of \(a_ia_j^{-1}\). Thus, \[ AE_{ij}A^{-1} = a_ia_j^{-1}E_{ij} \] This shows that each \(E_{ij}\) (with \(i \neq j\)) is an eigenvector for the conjugation action restricted to A. (c) Conjugation representation irreducibility

Step 1: Differentiate the conjugation action with respect to t

We have the conjugation action on \(g \in G\), as follows: \[ x(t)Yx(t)^{-1} \] where \(x(t) = \exp(tX)\). We want to find the derivative of this action with respect to t, evaluated at t = 0. Differentiating with respect to t, we get \[\frac{d}{dt}(x(t)Yx(t)^{-1}) \]

Step 2: Evaluate the derivative at t = 0

Evaluated at t = 0, we have \[ \left.\left.\frac{d}{d t} x(t) Y x(t)^{-1}\right|_{t=0} = X Y - Y X =[ X, Y ] \]

Step 3: Show that if V is a non-zero subspace of g that is stable under the conjugation action, then V = g

(We will use the hint provided). Using the fact that the derivative we found equals to the Lie bracket of X and Y, we can deduce that if a subspace V of g is stable under the conjugation action, then g acts on V through the adjoint representation Ad(g) = gxg^{-1}, which sends a vector space to another vector space. Now, consider the matrices \(E_{ij}\) (with \(i \neq j\)) that we proved to be eigenvectors for the conjugation action restricted to A. Since they are eigenvectors, they will stay in the same space V after the conjugation action. In this case, we have \[ Ad(A)(E_{ij}) = \mathbf{c}(A)E_{ij} = a_ia_j^{-1}E_{ij} \] Also, the action on the matrices \(H_i\) will give \[Ad(A)(H_{i}) = \mathbf{c}(A)(E_{ii} - E_{i+1, i+1}) = (a_{i}^{2} - a_{i+1}^{2})H_i \] Since V is a subspace stable under the conjugation action of G, and the action on the eigenvectors does not change their spaces, V must contain the entire Lie algebra g. Therefore, the conjugation representation of G on g is irreducible.

Key Concepts

Conjugation Action

The concept of conjugation action in Lie algebras involves transforming matrices through a process known as conjugation. In mathematical terms, for a group element \(g\) and a matrix \(X\), the conjugation action is defined as \(\mathbf{c}(g) X = g X g^{-1}\). This is a crucial operation in the study of Lie algebras, as it allows us to understand how matrices change under group actions.

When we apply the conjugation action to a matrix basis element, such as \(E_{ij}\), within a group of diagonal matrices \(A\), we achieve a transformation that can sometimes be simplified into a scalar multiplication of the original element. Specifically, in this context, each \(E_{ij}\) is an eigenvector of the conjugation action applied via the group \(A\). This eigenvector property implies that under the conjugation action, the matrix transforms but retains its structural identity, modulated only by a scalar factor derived from the entries of the diagonal matrix \(A\).

Understanding conjugation action also necessitates comprehending its role in proving irreducibility of representations, as shown by its application in establishing that certain subspaces remain invariant under this action.

Field of Characteristic 0

A field of characteristic 0 is a field in which the sum \(1 + 1 + \cdots + 1\) (\(n\) times) never sums to zero unless \(n = 0\). This property significantly affects the structure and behavior of algebras defined over such fields.

In the context of matrix algebras and Lie algebras, having a field with characteristic 0 ensures that we can perform all standard arithmetic operations with matrices without encountering anomalies that might occur in fields with other characteristics. Such fields include the rational numbers \(\mathbb{Q}\), real numbers \(\mathbb{R}\), and complex numbers \(\mathbb{C}\).

This is crucial when working with the trace of matrices or the eigenvector properties under conjugation because the characteristic properties help maintain the semantics and consistency of these operations across different mathematical contexts.

Matrix Basis

A matrix basis refers to a set of matrices that linearly span a given vector space, and are linearly independent, much like a vector basis in vector spaces. For the Lie algebra \(g\) defined over a field \(F\), which consists of matrices with a trace of zero, finding a proper matrix basis is critical for performing various algebraic operations and proving theoretical properties.

In this case, the basis consists of off-diagonal elements \(E_{ij}\) for \(i eq j\) and diagonal difference elements \(H_i = E_{ii} - E_{i+1,i+1}\). These are chosen such that:

• They cover the entire space of matrices having a trace of zero.
• They are linearly independent, meaning none can be written as a combination of the others.
• They directly correspond to basic transformations that preserve the trace zero property.

By constructing a basis in this way, it becomes possible to represent any matrix in the Lie algebra \(g\) as a linear combination of these basis matrices, simplifying the study of transformations like conjugation and differentiations like when proving representations.

Trace

In linear algebra, the trace of a matrix is the sum of its diagonal elements. The trace plays a fundamental role in numerous mathematical concepts, including Lie algebra. The trace function is particularly important because it relates to the notion of invariants in transformations.

Here are some key properties of the trace function:

• \(\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B)\)
• \(\text{Tr}(AB) = \text{Tr}(BA)\)

The invariant nature of the trace under cyclic permutations of matrices (i.e., \(\text{Tr}(AB) = \text{Tr}(BA)\)) is especially useful in the study of Lie algebras. For instance, in this specific Lie algebra of traceless matrices \(g\), each foundational transformation maintains a zero trace, ensuring that they belong to \(g\).

The trace restriction (trace being zero here) is a powerful constraint that limits the vector space dimensionality while maintaining essential algebraic properties, suitable for constructing and analyzing substructures such as subspaces under conjugate actions.