[Paper Review] Free subalgebras of division algebras over uncountable fields

Jason P. Bell, D. Rogalski|arXiv (Cornell University)|Nov 30, 2011

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper establishes that division algebras over uncountable algebraically closed fields of characteristic zero contain a free k-algebra on two generators unless they are left algebraic over every maximal subfield. The key result shows that finitely generated domains of GK-dimension less than 3 that do not satisfy a polynomial identity must have quotient division rings containing such free subalgebras, resolving a major case of the free subalgebra conjecture in low GK-dimension settings.

ABSTRACT

We study the existence of free subalgebras in division algebras, and prove the following general result: if $A$ is a noetherian domain which is countably generated over an uncountable algebraically closed field $k$ of characteristic 0, then either the quotient division algebra of $A$ contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if $k$ is an uncountable algebraically closed field and $A$ is a finitely generated $k$-algebra that is a domain of GK-dimension strictly less than 3, then either $A$ satisfies a polynomial identity, or the quotient division algebra of $A$ contains a free $k$-algebra on two generators.

Motivation & Objective

• To resolve the free subalgebra conjecture for domains of GK-dimension less than 3 over uncountable algebraically closed fields.
• To establish a criterion for when a division algebra contains a free k-algebra on two generators using centralizers and algebraicity.
• To show that the absence of a free subalgebra implies algebraicity over every maximal subfield, linking to the Kurosh problem and PI conditions.
• To extend previous results on Ore extensions and Weyl algebras to broader classes of noetherian domains.
• To provide a clean, general condition under which quotient division rings of finitely generated domains contain free subalgebras, especially in low GK-dimension.

Proposed method

• Uses a criterion from prior work (Corollary 4.9) that a division algebra D contains a free k-algebra on two generators if it is not left or right algebraic over the centralizer of a nonzero element a.
• Applies the key technical tool: over an uncountable field, D contains a free subalgebra if and only if D(t) does, where t is a commutative indeterminate.
• Employs GK-dimension arguments to rule out algebraicity: if Q(A) were algebraic over k(x), then dim_k(U^{2n}) would grow too fast, contradicting GKdim < 3.
• Uses the fact that if D(t) is not algebraic over E(t) = C(x; D(t)), then D(t) must be finite-dimensional over a field of transcendence degree 2, hence PI, contradicting the non-PI assumption.
• Applies results on the Weyl algebra in positive characteristic: if xux⁻¹ = u+1, then the subalgebra generated by x and u is isomorphic to the Weyl algebra, implying Q(A) is finite-dimensional over Q(R), hence PI.
• Uses Lemma 3.5 to transfer algebraicity from D(t) to D, and Lemma 2.8 to relate centralizers in D and D(t).

Experimental results

Research questions

• RQ1 Under what conditions does a quotient division ring of a noetherian domain over an uncountable field contain a free k-algebra on two generators?
• RQ2 Can the free subalgebra conjecture be verified for domains of GK-dimension less than 3?
• RQ3 What is the role of algebraicity over centralizers and maximal subfields in obstructing the existence of free subalgebras?
• RQ4 In positive characteristic, when does the absence of a solution to xux⁻¹ = u+1 ensure that Q(A) is not PI?
• RQ5 How does the uncountability of the base field enable stronger transfer results between D and D(t)?

Key findings

• If A is a countably generated noetherian domain over an uncountable algebraically closed field k of characteristic 0, then Q(A) either contains a free k-algebra on two generators or is left algebraic over every maximal subfield.
• For finitely generated k-algebras A of GK-dimension strictly less than 3, if A does not satisfy a polynomial identity, then Q(A) contains a free k-algebra on two generators.
• In positive characteristic, the same conclusion holds provided there is no element u satisfying xux⁻¹ = u+1 for any transcendental x ∈ A.
• The Weyl algebra case is recovered as a special case: its quotient division ring contains a free subalgebra, consistent with Makar-Limanov’s original result.
• If a domain A of GK-dimension < 3 is not PI, then its quotient division ring cannot be algebraic over any maximal subfield, and thus must contain a free subalgebra.
• The proof shows that assuming algebraicity over k(x) leads to a contradiction in growth rate, as dim_k(U^{2n}) would grow at least as fast as (n+2 choose 2)·n, violating GKdim < 3.