[Paper Review] Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building

Naoki Yamatsu|arXiv (Cornell University)|Nov 25, 2015

Algebraic structures and combinatorial models54 citations

TL;DR

This paper provides a comprehensive reference for finite-dimensional Lie algebras and their representations up to rank 20, focusing on applications in grand unified theories (GUTs) in 4D and 5D spacetime. It compiles essential data—such as conjugacy classes, Dynkin indices, Casimir invariants, anomaly coefficients, branching rules, and projection matrices—for classical and exceptional Lie algebras, enabling systematic construction and anomaly analysis in unified model building.

ABSTRACT

We give information about finite-dimensional Lie algebras and their representations for model building in 4 and 5 dimensions; e.g., conjugacy classes, types of representations, Weyl dimensional formulas, Dynkin indices, quadratic Casimir invariants, anomaly coefficients, projection matrices, and branching rules of Lie algebras and their subalgebras up to rank-20. We show what kind of Lie algebras can be applied for grand unified theories in 4 and 5 dimensions.

Motivation & Objective

• To compile and extend existing knowledge on finite-dimensional Lie algebras and their representations for unified model building in high-energy physics.
• To address missing or limited information in prior works—particularly regarding anomaly coefficients, higher-order Casimir invariants, and projection matrices—especially for higher-rank algebras.
• To provide a systematic, accessible reference for particle physicists constructing grand unified theories (GUTs) in 4 and 5 dimensions.
• To include detailed data on maximal subalgebras, branching rules, and representation types (complex, real, pseudo-real) for algebras up to rank 20.
• To support anomaly-free model construction by compiling Dynkin indices, quadratic Casimir invariants, and anomaly coefficients for irreducible representations.

Proposed method

• Systematic compilation of conjugacy classes, representation types (complex, self-conjugate, real, pseudo-real), and Weyl dimension formulas for Lie algebras up to rank 20.
• Derivation and tabulation of Dynkin indices, quadratic Casimir invariants, and anomaly coefficients using established group-theoretic methods.
• Application of projection matrices to compute branching rules for subalgebras, particularly maximal regular and special subalgebras.
• Use of Dynkin’s theorems and Weyl group orbit techniques to derive generic projection matrices for classical Lie algebras.
• Integration of results from prior works—such as McKay & Patera (1981), Mckay et al. (1977), and Slansky (1981yr)—into a unified, accessible format.
• Incorporation of computational tools (e.g., LieART, Susyno, LiE) to validate and cross-check results, especially for tensor products and branching rules.

Experimental results

Research questions

• RQ1 Which finite-dimensional Lie algebras and their representations are suitable for constructing consistent grand unified theories in 4 and 5 dimensions?
• RQ2 How can Dynkin indices, Casimir invariants, and anomaly coefficients be systematically computed and tabulated for high-rank algebras?
• RQ3 What are the complete branching rules and projection matrices for maximal subalgebras of classical and exceptional Lie algebras up to rank 20?
• RQ4 How can representation types (complex, real, pseudo-real) be classified and cross-verified across different Lie algebras?
• RQ5 What are the minimal and maximal dimensional representations for each Lie algebra, and how do they relate to anomaly cancellation in gauge theories?

Key findings

• The paper provides a complete tabulation of conjugacy classes and representation types (complex, self-conjugate, real, pseudo-real) for all simple Lie algebras up to rank 20.
• It compiles Dynkin indices and quadratic Casimir invariants for fundamental representations, enabling direct use in renormalization group equation calculations.
• Anomaly coefficients are systematically computed and tabulated, allowing immediate assessment of anomaly-freeness in chiral gauge theories.
• Projection matrices for maximal subalgebras of classical and exceptional Lie algebras are derived and made available for branching rule computations.
• Branching rules for representations up to 5,000 dimensions in classical algebras and 10,000 dimensions in exceptional algebras are tabulated using projection matrices.
• The paper identifies and classifies all maximal subalgebras of $A_n$, $B_n$, $C_n$, $D_n$, and exceptional algebras, including $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$, up to rank 20.