[Paper Review] Fine gradings on $\mathfrak{e}_6$

Cristina Draper, Antonio Viruel|arXiv (Cornell University)|Jul 28, 2012

Advanced Topics in Algebra19 references8 citations

TL;DR

This paper classifies all 14 fine group gradings on the exceptional Lie algebra 𝔰𝔩(6) over an algebraically closed field of characteristic zero, using the structure of maximal abelian diagonalizable subgroups (MAD-groups) in the automorphism group. It provides explicit descriptions of each grading via their associated MAD-groups, types, and fixed subalgebra dimensions, establishing a complete classification through a combination of group-theoretic techniques and computational verification on the Weyl group and root system projections.

ABSTRACT

There are fourteen fine gradings on the exceptional Lie algebra $\frak e_6$ over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that they are all.

Motivation & Objective

• To complete the classification of fine group gradings on finite-dimensional semisimple Lie algebras, focusing on the missing case of the E-family, specifically 𝔰𝔩(6).
• To describe all fine gradings on 𝔰𝔩(6) up to equivalence, identifying their associated MAD-groups and structural invariants.
• To establish a systematic method for classifying fine gradings on exceptional Lie algebras by leveraging the Weyl group action and projections of automorphisms.
• To distinguish the 14 fine gradings using invariants such as the type of grading (multiplicities of graded components) and the dimension of the fixed subalgebra.

Proposed method

• The authors analyze the automorphism group of 𝔰𝔩(6) and project its action onto the Weyl group, reducing the complexity from 78×78 matrices to 6×6 matrices.
• They use the extended Weyl group (automorphism group of the root system) to classify conjugacy classes of automorphisms and identify possible MAD-groups.
• The classification relies on constructing quasitori from commuting automorphisms and their fixed-point subgroups, particularly using inner and outer automorphisms of order 2 and 3.
• Representatives of orbits under the Weyl group action are extracted from existing literature (e.g., [2]), enabling hand computation with minimal reliance on computer algebra.
• The method distinguishes gradings via the type (72,0,2), (60,9), etc.) and the dimension of the fixed subalgebra, which uniquely identifies each conjugacy class.
• The authors exploit known results on elementary p-groups and their centralizers in the complex case to reduce computational effort and generalize prior classifications from 𝔰𝔩(2) and 𝔰𝔩(4).

Experimental results

Research questions

• RQ1 How many fine group gradings exist on the exceptional Lie algebra 𝔰𝔩(6) over an algebraically closed field of characteristic zero?
• RQ2 What are the structural invariants (type and fixed subalgebra dimension) that distinguish the 14 fine gradings?
• RQ3 Which of these gradings arise from inner automorphisms, and which involve outer automorphisms?
• RQ4 How can the classification be achieved without direct computation on the full automorphism group, given the high dimension of 𝔰𝔩(6)?
• RQ5 Can the MAD-groups of fine gradings be systematically constructed from known automorphisms and their projections?

Key findings

• There are exactly 14 fine group gradings on 𝔰𝔩(6) up to equivalence, as stated in Theorem 1.
• The 14 gradings are fully described by their associated MAD-groups, which are isomorphic to various products of ℤ₃⁴, (𝔽*)²×ℤ₃², ℤ₂⁶, ℤ₄×ℤ₂⁴, and other combinations.
• The grading types are encoded as tuples such as (72,0,2), (60,9), (48,1,0,7), etc., indicating the number of homogeneous components of each dimension.
• The dimension of the fixed subalgebra Lₑ ranges from 0 to 6, with ℒₑ of dimension 6 occurring only for the grading associated to (𝔽*)⁶.
• The first five MAD-groups contain no outer automorphisms, while the remaining nine involve outer automorphisms, particularly from the outer automorphism group of order 2 and 3.
• The classification is achieved via a combination of group-theoretic techniques and Weyl group orbit analysis, with only two proofs relying on computer assistance.