[Paper Review] (Finite) presentations of Bi-Zassenhaus loop algebras

Giuseppe Jurman|arXiv (Cornell University)|Apr 9, 2010

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper establishes that Bi-Zassenhaus loop algebras are finitely presented modulo their second center, constructing an explicit finite presentation for a graded Lie algebra whose quotient over its second center is isomorphic to a Bi-Zassenhaus loop algebra. The proof relies on generalized Jacobi identities and machine-assisted cohomological computations using the p-Quotient Program, with all results independently verified without reliance on computational checks.

ABSTRACT

We prove that Bi-Zassenhaus loop algebras are finitely presented up to central and second central elements. In particular, we show an explicit finite presentation for a Lie algebra whose quotient over its second centre is isomorphic to a Bi-Zassenhaus loop algebra.

Motivation & Objective

• To establish that Bi-Zassenhaus loop algebras are finitely presented up to central and second central elements.
• To construct an explicit finite presentation for a graded Lie algebra M(g,h) such that M(g,h)/Z₂(M(g,h)) ≅ Bₗ(g,h).
• To extend the classification framework of maximal class Lie algebras to characteristic two, where Bi-Zassenhaus algebras replace AFS algebras.
• To demonstrate that the defining property of finite presentation up to central and second central elements, previously known for AFS algebras, also holds for Bi-Zassenhaus loop algebras.

Proposed method

• The construction of the Lie algebra M(g,h) is achieved via cohomological arguments, building a finitely presented graded Lie algebra.
• Generalized Jacobi identities in characteristic two are expanded using the element z = x + y to simplify commutator expressions.
• The p-Quotient Program is used for machine computations to derive relations, though all proofs are independent of these calculations.
• Lucas’ Theorem is applied to evaluate binomial coefficients modulo 2, enabling the simplification of complex commutator expansions.
• The algebra is analyzed through its sequence of two-step centralizers and constituent lengths, with parameters q = 2^h and g determining the structure.
• Relations are derived by expanding Jacobi identities in exponential form, particularly for elements in homogeneous components of the algebra.

Experimental results

Research questions

• RQ1 Can Bi-Zassenhaus loop algebras be finitely presented modulo their second center?
• RQ2 Is there a finite presentation for a graded Lie algebra whose quotient over its second center is isomorphic to a Bi-Zassenhaus loop algebra?
• RQ3 Do the structural properties of AFS algebras—specifically finite presentation up to central and second central elements—extend to Bi-Zassenhaus algebras in characteristic two?
• RQ4 What role do generalized Jacobi identities and binomial coefficient identities modulo 2 play in deriving the defining relations of these algebras?

Key findings

• The paper proves that for every Bi-Zassenhaus loop algebra Bₗ(g,h), there exists a finitely presented graded Lie algebra M(g,h) such that M(g,h)/Z₂(M(g,h)) ≅ Bₗ(g,h).
• The construction of M(g,h) is achieved through cohomological methods and verified via generalized Jacobi identities and p-adic binomial coefficient analysis.
• The relations in M(g,h) are derived by expanding generalized Jacobi identities using the element z = x + y, which simplifies nested commutators.
• Binomial coefficients modulo 2 are evaluated using Lucas’ Theorem, showing that certain sums are congruent to 1 mod 2, which validates key commutator identities.
• The proof shows that the centre of M(g,h) is infinite-dimensional, implying that Bₗ(g,h) itself is not finitely presented, consistent with Neumann’s group-theoretic result applied to Lie algebras.
• The result confirms that Bi-Zassenhaus loop algebras are uniquely determined by a suitable finite-dimensional quotient, extending the classification framework to characteristic two.