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Visualizing Lie Subalgebras using Root and Weight Diagrams

Tevian Dray (Oregon State Univ.) and Aaron Wangberg (Winona State Univ.)


1. Introduction

Lie algebras are classified using Dynkin diagrams, which encode the geometric structure of root and weight diagrams associated with an algebra. This paper begins with an introduction to Lie algebras, roots, and Dynkin diagrams. We then show how Dynkin diagrams define an algebra's root and weight diagrams, and provide examples showing this construction. In Section 3, we develop two methods to analyze subdiagrams. We then apply these methods to the exceptional Lie algebra \(F_4\), and describe the slight modifications needed in order to apply them to \(E_6\). We conclude by listing all Lie subalgebras of \(E_6\).

2. Root and Weight Diagrams of Lie Algebras

We summarize here some basic properties of root and weight diagrams. Further information can be found in [1], [2], and [3]. A description of how root and weight diagrams are applied to particle physics is also given in [2].

2.1 Lie Algebras

A Lie algebra \(g\) of dimension \(n\) is an \(n\)-dimensional vector space along with a product \([\ ,\ ]:g \times g \to g\), called a commutator, which is anti-commutative (\([x,y] = -[y,x]\)) and satisfies the Jacobi Identity $$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$$ for all \(x,y,z \in g\). A Lie algebra is called simple if it is non-abelian and contains no non-trivial ideals. All complex semi-simple Lie algebras are the direct sum of simple Lie algebras. Thus, we follow the standard practice of studying the simple algebras, which are the building blocks of the semi-simple algebras.

There are four infinite families of Lie algebras as well as five exceptional Lie algebras. The (compact, real forms of) the algebras in the four infinite families correspond to special unitary matrices (or their generalizations) over different division algebras. The algebras \(B_n\) and \(D_n\) correspond in this way to real special orthogonal groups in odd and even dimensions, \(SO(2n+1)\) and \(SO(2n)\), respectively. The algebras \(A_n\) correspond to complex special unitary groups \(SU(n+1)\), and the algebras \(C_n\) correspond to unitary groups \(SU(n,\mathbb{H})\) over the quaternions, which are more usually described as (some of the) symplectic groups.

While Lie algebras are usually classified using their complex representations, there are particular real representations, based upon the division algebras, which are of interest in particle physics. Manogue and Schray [4] describe the use of quaternions \(\mathbb{H}\) to construct \(su(2,\mathbb{H})\) and \(sl(2, \mathbb{H})\), which are real representations of \(B_2=so(5)\) and \(D_3 = so(5,1)\), respectively. As they discuss, their construction naturally generalizes to the octonions \(\mathbb{O}\), yielding the real representations \(su(2,\mathbb{O})\) and \(sl(2,\mathbb{O})\) of \(B_4 = so(9)\) and \(D_5 = so(9,1)\), respectively. This can be further generalized to the \(3 \times 3\) case, resulting in \(su(3,\mathbb{O})\) and \(sl(3, \mathbb{O})\), which preserve the trace and determinant, respectively, of a \(3 \times 3\) octonionic hermitian matrix, and which are real representations of two of the exceptional Lie algebras, namely \(F_4\) and \(E_6\), respectively [5]. The remaining three exceptional Lie algebras are also related to the octonions [6, 7]. The smallest, \(G_2\), preserves the octonionic multiplication table and is 14-dimensional, while \(E_7\) and \(E_8\) have dimensions 133 and 248, respectively. A major stem in describing the infinite-dimensional unitary representation of the split form of \(E_8\) was recently completed by the Atlas Project [8].

In sections 2 and 3, we label the Lie algebras using their standard name (e.g. \(A_n\), \(B_n\)) and also with a standard complex representation (e.g. \(su(3)\), (so(7))). In section 4, when discussing the subalgebras of \(E_6\), we also give particular choices of real representations.

2.2 Roots and Root Diagrams

Every simple Lie algebra \(g\) contains a Cartan subalgebra \(h \subset g\), whose dimension is called the rank of \(g\). The Cartan subalgebra \(h\) is a maximal abelian subalgebra such that \(ad H\) is diagonalizable for all \(H \in h\). The Killing form can be used to choose an orthonormal basis \(\{h_{1}, \cdots, h_{l}\}\) of \(h\) which can be extended to a basis \[\{ h_{1}, \cdots, h_{l}, g_1, g_{-1}, g_2, g_{-2},\cdots, g_{\frac{n-l}{2}}, g_{-\frac{n-l}{2}} \} \] of \(g\) satisfying:

  1. \([h_{i}, g_{j}] = \lambda_i^j g_{j}\) (no sum), \(\lambda_i^j \in \mathbb{R}\)
  2. \([h_{i}, h_{j}] = 0 \)
  3. \([g_{j}, g_{-j}] \in h\)

The basis elements \(g_j\) and \(g_{-j}\) are referred to as raising and lowering operators. Property 1 associates every \(g_{j}\) with an \(l\)-tuple of real numbers \(r^{j} = \langle\lambda_1^j, \cdots, \lambda_l^j\rangle\), called roots of the algebra, and this association is one-to-one. Further, if \(r^{j}\) is a root, then so is \(-r^{j} = r^{-j}\), and these are the only two real multiples of \(r^{j}\) which are roots. According to Property 2, each \(h_i\) is associated with the \(l\)-tuple \(\langle0, \cdots, 0\rangle\). Because this association holds for every \(h_i \in h\), these \(l\)-tuples are sometimes referred to as zero roots. For raising and lowering operators \(g_j\) and \(g_{-j}\), Property 3 states that \(r^{j} + r^{-j} = \langle0, \cdots, 0\rangle\).

Let \(\Delta\) denote the collection of non-zero roots. For roots \(r^{i}\) and \(r^{j} \ne -r^i\), if there exists \(r^{k} \in \Delta\) such that \(r^i + r^j = r^k\), then the associated operators for \(r^i\) and \(r^j\) do not commute, that is, \([ g_i, g_j ] \ne 0\). In this case, \([g_i, g_j] = C^{k}_{ij}g_{k}\) (no sum), with \(C^k_{ij} \in \mathbb{C}, C^i_{ij} \ne 0\). If \(r^i + r^j \not\in \Delta\), then \([g_i, g_j] = 0\).

When plotted in \(\mathbb{R}^l\), the set of roots provide a geometric description of the algebra. Each root is associated with a vector in \(\mathbb{R}^l\). We draw \(l\) zero vectors at the origin for the \(l\) zero roots corresponding to the basis \(h_1, \cdots, h_l\) of the Cartan subalgebra. For the time being, we then plot each non-zero root \(r^i = \langle\lambda_1^i, \cdots, \lambda_l^i\rangle\) as a vector extending from the origin to the point \(\langle\lambda_1^i, \cdots, \lambda_l^i\rangle\). The terminal point of each root vector is called a state. As is commonly done, we use \(r^i\) to refer to both the root vector and the state. In addition, we allow translations of the root vectors to start at any state, and connect two states \(r^i\) and \(r^j\) by the root vector \(r^k\) when \(r^k + r^i = r^j\) in the root system. The resulting diagram is called a root diagram.

As an example, consider the algebra \(su(2)\), which is classified as \(A_1\). The algebra \(su(2)\) is the set of \(2 \times 2\) complex traceless Hermitian matrices. Setting

\[ \sigma_1 = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right] \]   \[ \sigma_2 = \left[ \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right] \]   \[ \sigma_3 = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right] \]

we choose the basis \(h_1 = \frac{1}{2}\sigma_3\) for the Cartan subalgebra \(h\), and use \(g_1 = \frac{1}{2}(\sigma_1+i\sigma_2)\) and \(g_{-1} = \frac{1}{2}(\sigma_1-i\sigma_2)\) to extend this basis for all of \(su(2)\). Then

  1. \([h_1, h_1] = 0\)
  2. \([h_1, g_1] = 1 g_1\)
  3. \([h_1, g_{-1}] = -1 g_{-1}\)
  4. \([g_{1}, g_{-1}] = h_1\)

By Properties 2 and 3, we associate the root vector \(r^1 = \langle1\rangle\) with the raising operator \(g_1\) and the root vector \(r^{-1} = \langle-1\rangle\) with the lowering operator \(g_{-1}\). Using the zero root \(\langle0\rangle\) associated with \(h_1\), we plot the corresponding three points \((1)\), \((-1)\), and \((0)\) for the states \(r^1\), \(r^{-1}\), and \(h_1\). We then connect the states using the root vectors. Instead of displaying both root vectors \(r^1\) and \(r^{-1}\) extending from the origin, we have chosen to use only the root vector \(r^{-1}\), as \(r^{-1} = - r^{1}\), to connect the states \((1)\) and \((0)\) to the states \((0)\) and \((-1)\), respectively. The resulting root diagram is illustrated in Figure 1.

 Figure 1. Root Diagram of \(A_1 = su(2)\)

Tevian Dray (Oregon State Univ.) and Aaron Wangberg (Winona State Univ.), "Visualizing Lie Subalgebras using Root and Weight Diagrams," Convergence (February 2010), DOI:10.4169/loci003287