Question

Find the Cartan metrics for \(\mathfrak{o}(3,1)\) and \(\mathfrak{p}(3,1)\), and show directly that the first is semisimple but the second is not.

Short answer

The Cartan metric for \(\mathfrak{o}(3,1)\) is \(-1/2\) times the identity matrix, and is semisimple. While for \(\mathfrak{p}(3,1)\), the Cartan's metric is more complex and non-invariant, hence it is not semisimple.

Step-by-step solution

Cartan's Metric for \(\mathfrak{o}(3,1)\)

Start by understanding that \(\mathfrak{o}(3,1)\) is essentially the Lorentz Algebra. Remember, the Cartan metric is defined as \(h_{ab} = tr(t_a t_b)\), where \(t_a\) and \(t_b\) are the generators of the Lie algebra under consideration. This is a constant matrix and it turns out that for \(\mathfrak{o}(3,1)\), the Cartan's metric is \(-1/2\) times the identity matrix, say \(I\). So mathematically, for every \(t_a, t_b\) in \(\mathfrak{o}(3,1)\) we have \(h_{ab} = -1/2 tr(t_at_b)\). Exploring further, we find that this metric is invariant under the adjoint action.

Verifying Semisimplicity for \(\mathfrak{o}(3,1)\)

An algebra \(\mathfrak{g}\) is said to be semisimple if its killing form, \(B(X, Y) = tr(ad(X)ad(Y))\), is non-degenerate. Remember that, the Cartan-Killing form is actually proportional to the Cartan metric for a semisimple algebra. Our Cartan metric for \(\mathfrak{o}(3,1)\) is non-degenerate. Therefore, it satisfies one of the conditions of semisimplicity. Additionally, the Lie algebra \(\mathfrak{o}(3,1)\) has no Abelian ideals, and thus is semisimple.

Cartan's Metric for \(\mathfrak{p}(3,1)\)

\(\mathfrak{p}(3,1)\) (or the Poincare Algebra) is more complex. The generators for \(\mathfrak{p}(3,1)\) consist of those of \(\mathfrak{o}(3,1)\), plus additional generators. Due to the additional complexity, the Cartan's metric for \(\mathfrak{p}(3,1)\) is not invariant under the adjoint action.

Verifying Non-Semisimplicity for \(\mathfrak{p}(3,1)\)

By definition, a semisimple Lie algebra has a non-degenerate Killing form. In our case, for \(\mathfrak{p}(3,1)\), we find that the Killing form is degenerate, because the Cartan metric is not invariant under the adjoint action. Hence, \(\mathfrak{p}(3,1)\) is not considered semisimple.

Key concepts

Lorentz Algebra

The Lorentz Algebra, denoted as \( \mathfrak{o}(3,1) \), is fundamental in our understanding of the symmetries of spacetime in relativity. It comprises generators that correspond to rotations and boosts, which are transformations connecting different inertial frames in a spacetime with three spatial dimensions and one time dimension. These generators satisfy specific commutation relations that form the structure of this Lie algebra.

In terms of physics, the Lorentz Algebra underpins the behavior of particles and fields when observed from different reference frames, ensuring that the laws of physics are the same for all observers. Its mathematical properties are critical because they encode the geometric structure of spacetime itself.

Lie algebra

A Lie algebra is a mathematical structure used extensively in areas of physics and geometry to describe symmetries. It consists of elements (such as matrices, or more abstract entities) that can be added together and scaled by numbers (fields). What primarily characterizes a Lie algebra are the commutators — the specific way its elements combine with one another, typically denoted \( [X, Y] \).

The power of Lie algebras lies in their ability to provide a formal framework for studying continuous transformation groups, known as Lie groups. Through this, they become invaluable tools for understanding the underlying symmetries of various physical systems, and much of modern theoretical physics is built upon their foundations. They serve as the linchpin connecting algebra with geometry and physics.

Killing form

The Killing form, named after mathematician Wilhelm Killing, is a bilinear form used to determine many important properties of a Lie algebra. Given two elements \(X \text{ and } Y\) of a Lie algebra, it is defined as \(B(X, Y) = \text{tr}(\text{ad}(X)\text{ad}(Y))\), where \(\text{ad}\) is the adjoint action of the Lie algebra.

Significance of the Killing form

It serves as a tool to classify Lie algebras because it captures intrinsic geometric properties. For instance, a Lie algebra is semisimple if the Killing form is non-degenerate, meaning that if \(B(X, Y) = 0\) for all \(Y\), then \(X\) must be the zero element. Its non-degeneracy is a central feature that enforces the robustness of the structure and symmetries described by the algebra.

Adjoint action

The adjoint action is an operation that describes how Lie algebra elements (such as matrices or differential operators) act upon each other from within their own structure. When an element \(X\) of a Lie algebra is used to \(\text{ad}\)-act on another element \(Y\), the result is the Lie bracket of \(X \text{ and } Y\), \(\text{ad}(X)(Y) = [X, Y]\).

The adjoint action has deep implications in the study of a Lie algebra's structure. It tells us how the algebra's generators (like angular momentum or charge operators in physics) influence each other, and thus, it is critical for understanding the symmetry operations that the algebra represents.

Invariants and Stability

One of the remarkable features of the adjoint action is its role in defining invariants, which are properties that remain unchanged under the action. For a semisimple Lie algebra, the Killing form is an example of such an invariant under the adjoint action, which directly links to the concept of semisimplicity and informs us whether the algebra has a certain rigidity and consistency in its structure.