Cartan Subalgebras, Roots, and Weights in su(2)

Welcome to the first lesson of our new module on advanced Lie algebras. In the previous module, we concluded by examining isospin, a powerful application of SU(2) symmetry that brought order to the chaotic world of hadrons. You saw how the familiar mathematics of angular momentum could be repurposed to describe an internal, abstract property of particles, leading to concrete physical predictions.

The success of isospin was a clue that an even deeper structure might lie beneath. To uncover it, physicists needed to generalize from SU(2) to more complex groups like SU(3). Doing so requires a more powerful and systematic framework for analyzing the structure of any Lie algebra and its representations. This lesson introduces the essential tools for that analysis: the Cartan subalgebra, roots, and weights. We will define these abstract concepts and then ground them in the familiar territory of the su(2) algebra, constructing its root and weight diagrams. This will lay the groundwork for understanding the beautiful geometric patterns that emerge in higher-rank algebras.

A General Strategy for Representations

Our study of su(2) representations was guided by the operator $J_{z}$ . We classified irreducible representations by a highest weight $j$ , and all states within that representation were labeled by their $J_{z}$ eigenvalue, $m_{j}$ . The raising and lowering operators, $J_{+}$ and $J_{-}$ , then allowed us to move between these states.

This provides a template for a general strategy:

For a given Lie algebra, find a maximal set of commuting generators.
Use the simultaneous eigenvalues of these generators to label the basis states of a representation.
The remaining generators of the algebra will then act as raising or lowering operators, moving states between different sets of eigenvalues.

The maximal set of commuting generators is called the Cartan subalgebra, and it is the cornerstone of the classification scheme for semisimple Lie algebras. To formalize this, we first need to introduce a special representation that every Lie algebra has for free.

The Adjoint Representation

A Lie algebra $g$ is, first and foremost, a vector space. What if we use the algebra itself as the vector space for a representation? This canonical representation is called the adjoint representation.

An element $X \in g$ acts on another element $Y \in g$ via the Lie bracket. We define the linear map $ad_{X}$ as:

ad_{X} (Y) = [X, Y]

The map $X \mapsto ad_{X}$ is the adjoint representation of the Lie algebra $g$ . Each $ad_{X}$ is a linear operator (a matrix, in a given basis) acting on the vector space $g$ .

The following video provides a clear definition of the adjoint representation for both a Lie group and its algebra, and derives the crucial result that the Lie algebra representation is simply the commutator.

Lie groups and Lie algebras: The adjoint representation

This video, "Lie groups and Lie algebras: The adjoint representation" by Jonathan Evans, introduces the concept and calculates the induced map for the Lie algebra.

Please watch from the beginning up to the derivation. The key takeaway is the calculation starting around 4:28, which shows that for the Lie algebra, the adjoint action of an element X on an element Y is simply their Lie bracket, [X, Y].

Cartan Subalgebra, Roots, and Weights

With the adjoint representation in hand, we can now give more precise definitions.

Lie Groups and Lie Algebras for Physicists

This section from Harold Steinacker's lecture notes, "Lie Groups and Lie Algebras for Physicists," provides excellent, physics-oriented definitions of these core concepts.

Please read the subsection 8.1 Cartan subalgebras. This text will formally define: The Cartan subalgebra \mathfrak{g}_0 (or \mathfrak{h}), a maximal abelian subalgebra whose elements H \in \mathfrak{h} are diagonalizable in the adjoint representation. Its dimension is the rank of the algebra. The roots \alpha, which are linear functionals providing the eigenvalues in the adjoint representation: \text{ad}_H(X_\alpha) = [H, X_\alpha] = \alpha(H) X_\alpha. The root space decomposition, which breaks the algebra into the Cartan subalgebra (the zero-root space) and the spaces spanned by the root vectors X_\alpha.

To summarize and connect these ideas:

The Cartan subalgebra $h$ is our chosen set of simultaneously diagonalizable generators. For a rank- $l$ algebra, we choose a basis ${H_{1}, \dots, H_{l}}$ .
The adjoint representation provides a matrix for each generator, $ad_{H_{i}}$ . We can simultaneously diagonalize these matrices.
The other basis vectors of the algebra, let's call them $X_{α}$ , are the eigenvectors. They satisfy $[H_{i}, X_{α}] = α_{i} X_{α}$ for each $H_{i}$ .
The tuple of eigenvalues $(α_{1}, \dots, α_{l})$ is a root, often denoted by the vector $α$ . Roots describe the internal structure of the Lie algebra itself.

Now, we extend this from the adjoint representation to any representation.

Lie Groups and Lie Algebras for Physicists

This subsequent section of Steinacker's notes defines weights for an arbitrary representation.

Please read the subsection 8.2 Useful concepts for representations. This introduces: Weights \lambda, which are the simultaneous eigenvalues of the Cartan generators in an arbitrary representation V. A vector |v_\lambda\rangle \in V with weight \lambda satisfies \pi(H)|v_\lambda\rangle = \lambda(H)|v_\lambda\rangle. The action of root vectors X_\alpha as raising/lowering operators, which shift the weight of a state by the root \alpha.

The most important connection is this: the roots of a Lie algebra are simply the non-zero weights of its adjoint representation.

Example: The Structure of `su(2)`

Let's make this concrete with the su(2) algebra, whose generators you know well from quantum mechanics. We will use the physics basis ${J_{1}, J_{2}, J_{3}}$ satisfying $[J_{i}, J_{j}] = i ϵ_{ij k} J_{k}$ (with $ℏ = 1$ ), or more conveniently, the basis ${J_{3}, J_{+}, J_{-}}$ where $J_{\pm} = J_{1} \pm i J_{2}$ . The commutation relations are:

[J_{3}, J_{\pm}] = \pm J_{\pm}, [J_{+}, J_{-}] = 2 J_{3}

Cartan Subalgebra: We need a maximal set of commuting generators. Here, it's clear that only one generator can be chosen, typically $J_{3}$ . Thus, the Cartan subalgebra is $h = span {J_{3}}$ . The algebra su(2) has rank 1.
Roots: To find the roots, we must find the eigenvalues of $ad_{J_{3}}$ . We let $ad_{J_{3}}$ act on all the basis vectors of the algebra ${J_{3}, J_{+}, J_{-}}$ .
- $ad_{J_{3}} (J_{3}) = [J_{3}, J_{3}] = 0$ . The eigenvalue is 0. This corresponds to the "zero root," and the eigenvector is the Cartan element itself.
- $ad_{J_{3}} (J_{+}) = [J_{3}, J_{+}] = (+ 1) J_{+}$ . The eigenvalue is +1. This gives us a root $α_{+} = 1$ .
- $ad_{J_{3}} (J_{-}) = [J_{3}, J_{-}] = (- 1) J_{-}$ . The eigenvalue is -1. This gives us a root $α_{-} = - 1$ .
The set of non-zero roots is ${- 1, + 1}$ .
Root Diagram: Since the rank is 1, the roots are single numbers and can be plotted on a 1D line. The root diagram consists of points at the locations of the roots. Including the zero root for the Cartan element, we get:

Visualizing Lie Subalgebras using Root and Weight Diagrams

This is the root diagram for su(2), also known as the A1 algebra. The dots at -1 and +1 represent the non-zero roots, corresponding to the lowering and raising operators J- and J+. The dot at 0 represents the zero root, corresponding to the Cartan generator J3.

Weights and Weight Diagrams: Now consider arbitrary irreducible representations of su(2), which are labeled by spin $j$ .
- Spin-1/2 (Fundamental Rep): The representation space is spanned by states $∣ j, m_{j} ⟩$ . Here, $j = 1/2$ , and the states are $∣1/2, + 1/2 ⟩$ and $∣1/2, - 1/2 ⟩$ . The weights are the eigenvalues of $J_{3}$ , which are $w = {+ 1/2, - 1/2}$ .
- Spin-1 (Adjoint Rep): Here, $j = 1$ . The states are $∣1, + 1 ⟩, ∣1, 0 ⟩, ∣1, - 1 ⟩$ . The weights are $w = {+ 1, 0, - 1}$ .
- Spin-j (General Rep): The weights are $w = {- j, - j + 1, \dots, j - 1, j}$ .
The weight diagram for a representation is simply a plot of its weights. For su(2), these are just sets of points on a line. Notice that the weight diagram for the spin-1 representation is identical to the root diagram. This is a general feature.

The action of the root vectors $J_{+}$ (with root +1) and $J_{-}$ (with root -1) is to shift the weights, as expected:

$J_{+}$ acting on a state with weight $m_{j}$ produces a state with weight $m_{j} + 1$ .
$J_{-}$ acting on a state with weight $m_{j}$ produces a state with weight $m_{j} - 1$ .

This confirms the general principle: $X_{α}$ shifts a weight $w$ to $w + α$ .

The following resource provides a concise summary of this entire process for su(2).

Visualizing Lie Subalgebras using Root and Weight Diagrams

This webpage by Tevian Dray gives a clear, visual introduction to these concepts.

Please read the following three short sections: Section 2.2 defines the Cartan subalgebra and roots, and how to construct a root diagram. Section 2.2's example walks through the explicit construction for su(2) that we just discussed. Section 2.3 defines weights and weight diagrams, and explicitly states the connection between roots and weights.

Conclusion

In this lesson, you have learned the fundamental strategy for classifying the representations of any semisimple Lie algebra. This machinery, while appearing somewhat abstract at first, provides a powerful geometric visualization of the algebra's structure.

Key Takeaways:

The Cartan subalgebra is a maximal set of commuting generators ( $H_{i}$ ) whose eigenvalues are used to label states. Its dimension is the rank of the algebra.
The adjoint representation describes how generators of the algebra act on each other via the commutator, $ad_{X} (Y) = [X, Y]$ .
Roots ( $α$ ) are the non-zero eigenvalue vectors of the Cartan generators in the adjoint representation. The diagram of these vectors is the root diagram, which maps out the structure of the algebra.
Weights ( $w$ ) are the eigenvalue vectors of the Cartan generators in any given representation. The diagram of these vectors is the weight diagram for that representation.
The root vectors $X_{α}$ act as generalized raising and lowering operators, shifting states between weights: $X_{α} ∣ w ⟩ \propto ∣ w + α ⟩$ .

For su(2), a rank-1 algebra, these diagrams are simple one-dimensional patterns. The real power of this formalism will become clear in our next lesson, where we apply it to the rank-2 algebra su(3). You will see how the root and weight diagrams blossom into two-dimensional geometric figures—hexagons and triangles—that directly correspond to the isospin-hypercharge plots of the Eightfold Way, bringing a profound geometric order to the particle zoo.

Recommended resources

A General Strategy for Representations

The Adjoint Representation

Cartan Subalgebra, Roots, and Weights

Example: The Structure of su(2)

Conclusion

Example: The Structure of `su(2)`