[Paper Review] Enveloping algebras of solvable Malcev algebras of dimension five

Murray R. Bremner, Marina Tvalavadze|arXiv (Cornell University)|Aug 11, 2010

Advanced Topics in Algebra2 references3 citations

TL;DR

This paper constructs explicit structure constants for the universal nonassociative and alternative enveloping algebras of a one-parameter family of 5-dimensional solvable Malcev algebras. It determines the center of the universal enveloping algebra, showing it is non-trivial precisely when the parameter γ is rational, and proves these Malcev algebras are special by embedding them into alternative algebras.

ABSTRACT

We study the universal enveloping algebras of the one-parameter family of solvable 5-dimensional non-Lie Malcev algebras. We explicitly determine the universal nonassociative enveloping algebras (in the sense of Perez-Izquierdo and Shestakov) and the centers of the universal enveloping algebras. We also determine the universal alternative enveloping algebras.

Motivation & Objective

• To compute explicit structure constants for the universal nonassociative enveloping algebra $ U(M) $ and the universal alternative enveloping algebra $ A(M) $ of a 5-dimensional solvable Malcev algebra.
• To determine the center of the universal enveloping algebra $ U(M) $, identifying conditions under which it is non-trivial.
• To prove that the one-parameter family of 5D solvable Malcev algebras is special, i.e., embeddable into the commutator algebra of an alternative algebra.
• To construct a smaller, infinite-dimensional alternative algebra $ \mathbb{A}_\gamma $ that contains the Malcev algebra as a subalgebra of its commutator algebra.
• To investigate whether finite-dimensional special Malcev algebras necessarily admit finite-dimensional alternative envelopes, posing an open problem in the field.

Proposed method

• Construct $ U(M) $ as the quotient of the free nonassociative algebra on a basis of $ M $ by the ideal generated by relations encoding Malcev identities and the commutator structure.
• Use left-tapped monomials $ \overline{a}_I $ indexed by non-decreasing multi-indices to form a basis of $ U(M) $, ensuring PBW-type isomorphism with the polynomial algebra on $ M $.
• Derive recursive formulas (Lemma 2.3) for computing commutators $ [x,s] $ and left multiplication $ sx $ in $ U(M) $, using the structure constants of $ M $ and associator identities.
• Define the alternator ideal $ I(M) $ as the ideal generated by all associators $ (x,y,z) $, and construct $ A(M) = U(M)/I(M) $ as the universal alternative quotient.
• Construct an explicit alternative algebra $ \mathbb{A}_\gamma $ with basis $ \{a^r, b, c, d, e \mid r \geq 1\} $, defining products via $ a^r \cdot a^s = a^{r+s} $, and $ e \cdot a^r = (-\gamma)^r e $, and verify its alternativity via associator vanishing.
• Use quotient arguments modulo $ \mathbb{I}_\gamma = \mathrm{span}\{a^t - a^s\} $ to show that $ \mathbb{A}_\gamma' / \mathbb{I}_\gamma $ is alternative, and extend this to the full $ \mathbb{A}_\gamma $ by checking all basis associators vanish.

Experimental results

Research questions

• RQ1 What are the explicit structure constants for the universal nonassociative enveloping algebra $ U(M) $ of the 5D solvable Malcev algebra $ \mathbb{M}_\gamma $?
• RQ2 What is the structure of the center of $ U(M) $, and for which values of $ \gamma $ is it non-trivial?
• RQ3 Is the universal alternative enveloping algebra $ A(M) $ finite-dimensional or infinite-dimensional?
• RQ4 Can the Malcev algebra $ \mathbb{M}_\gamma $ be embedded into the commutator algebra of a finite-dimensional alternative algebra?
• RQ5 Does every finite-dimensional special Malcev algebra admit a finite-dimensional alternative enveloping algebra?

Key findings

• The universal nonassociative enveloping algebra $ U(M) $ of the 5D solvable Malcev algebra $ \mathbb{M}_\gamma $ has an explicit basis of left-tapped monomials and structure constants derived from recursive commutator formulas.
• The center of $ U(M) $ is non-trivial if and only if the parameter $ \gamma $ is rational, with explicit central elements constructed via associator identities.
• The universal alternative enveloping algebra $ A(M) $ is infinite-dimensional, and is realized as the quotient $ U(M)/I(M) $, where $ I(M) $ is the alternator ideal.
• The Malcev algebra $ \mathbb{M}_\gamma $ is special: it embeds as a subalgebra of the commutator algebra of the infinite-dimensional alternative algebra $ \mathbb{A}_\gamma $.
• The algebra $ \mathbb{A}_\gamma $ is alternative, as all basis associators vanish, verified by direct computation and quotient arguments modulo $ \mathbb{I}_\gamma $.
• The construction of $ \mathbb{A}_\gamma $ provides a smaller, infinite-dimensional alternative envelope than $ U(M) $, and the alternativity is confirmed by showing $ (x,y,z) = (y,x,z) $ and $ (x,y,z) = (x,z,y) $ for all basis elements.