[Paper Review] A simple E8 construction

Sergio L. Cacciatori, Francesco Dalla Piazza+1 more|arXiv (Cornell University)|Jul 16, 2012

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper completes a program to construct an explicit generalized Euler parametrization for the exceptional Lie group $E_8$, using its maximal subgroup $\mathrm{Ss}(16) = \mathrm{Spin}(16)/\mathbb{Z}_2$. By leveraging a Clifford algebra realization of the $E_8$ Lie algebra via $\mathrm{Spin}(16)$ and its 128-dimensional spinor representation, the authors derive a parameterization with precisely defined ranges for the $y^a$ parameters that ensure global, non-redundant coverage of $E_8$, resolving a longstanding challenge in numerical and theoretical physics applications.

ABSTRACT

In this short letter we conclude our program, started in [J. Math. Phys. 46 (2005) 083512], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. In this last step we solve the problem for E8.

Motivation & Objective

• To complete the program of explicit generalized Euler parametrizations for all compact simply connected exceptional Lie groups, including the previously unresolved case of $E_8$.
• To provide a globally non-redundant parametrization of $E_8$ by specifying exact ranges for the parameters, avoiding numerical complications in simulations.
• To extend the method developed in prior works to $E_8$ using the maximal subgroup $\mathrm{Ss}(16)$ and the $\mathbf{128}$ spinor representation of $\mathrm{Spin}(16)$.
• To resolve the long-standing challenge of determining the correct parameter ranges for $E_8$ in generalized Euler coordinates, ensuring one-to-one coverage of the group manifold.

Proposed method

• Realize the $E_8$ Lie algebra using the compact real form of $\mathrm{Spin}(16)$ and its 128-dimensional Majorana-Weyl spinor representation $\Delta_+$.
• Construct the adjoint representation $\mathbf{248}$ of $E_8$ by combining the $\mathrm{Spin}(16)$ generators $J_{ij}$ with the spinor generators $Q_\alpha$, using commutator relations involving $\gamma$-matrices.
• Define the group $E_8$ via the parametrization $E_8[x; y; z] = (S[x]/\mathbb{Z}_2^8) \exp(\sum_a y^a C_a) S[z]$, where $S$ is a parametrization of $\mathrm{Ss}(16)$.
• Implement the $\mathbb{Z}_2^8$ quotient by restricting the ranges of 8 parameters in $x$ to half-periods, eliminating redundancies.
• Determine the injective range for the $y^a$ parameters using the simple roots $\alpha^i$ and the longest root $\alpha^L$, imposing constraints $0 \leq \sum_a \alpha^i_a y^a < \pi$ for $i=1,\dots,8$ and $0 \leq \sum_a \alpha^L_a y^a < \pi$.
• Explicitly compute the parameter bounds by substituting the chosen simple roots and longest root $\alpha^L = (0,0,0,0,0,0,1,1)$, yielding nested inequalities for $y^1$ through $y^8$.

Experimental results

Research questions

• RQ1 How can a globally non-redundant generalized Euler parametrization be constructed for the exceptional Lie group $E_8$?
• RQ2 What are the exact parameter ranges for the Cartan subalgebra generators that ensure one-to-one coverage of $E_8$?
• RQ3 How can the $\mathbb{Z}_2^8$ quotient in the maximal subgroup $\mathrm{Ss}(16)$ be systematically implemented in the parametrization to remove redundancies?
• RQ4 What is the explicit realization of the $E_8$ Lie algebra using $\mathrm{Spin}(16)$ and its spinor representation?
• RQ5 How do the simple roots and the longest root of $E_8$ constrain the parameter space in the generalized Euler construction?

Key findings

• The paper provides a complete, globally injective parametrization of $E_8$ using the generalized Euler method, with the $y^a$ parameters constrained by 8 inequalities derived from the simple roots and one from the longest root.
• The parameter range for $y^1$ is restricted to $0 \leq y^1 < \pi/6$, ensuring the first parameter covers only a sixth of the full period.
• The ranges for $y^2$ through $y^8$ are successively nested and depend on the previous parameters, with $y^8$ bounded by a linear inequality involving all prior $y^a$ and $y^8$ itself.
• The longest root $\alpha^L = (0,0,0,0,0,0,1,1)$ leads to the constraint $0 \leq y^7 + y^8 < \pi$, which is critical for injectivity.
• The method ensures that the entire $E_8$ group is covered exactly once (up to a set of measure zero), resolving a key challenge in numerical simulations.
• The construction is explicitly realizable using a Mathematica program available at http://www.dfm.uninsubria.it/E8/, enabling computational verification of the parametrization and Cartan subalgebra elements.