Makar-Limanov's symmetric free subalgebra conjecture for division rings with involution

Let DD be a division ring with center kk, and let \ast be a kk-involution on DD, meaning a kk-linear map satisfying (ab)=ba(ab)^{\ast}=b^{\ast}a^{\ast} and a=aa^{\ast\ast}=a for all a,bDa,b\in D. An element aDa\in D is symmetric with respect to \ast when a=aa^{\ast}=a.

Makar-Limanov's involutional conjecture. If DD is infinite dimensional over kk and finitely generated as a division kk-algebra, then there exist two symmetric elements in DD which freely generate a free kk-subalgebra of DD.

This is presented as an involutional version of Makar-Limanov's conjecture. The paper describes its results as supporting evidence, so the conjecture remains open in general.

Sources & referencesView supporting material

Primary source

Vitor O. Ferreira, Érica Z. Fornaroli and Jairo Z. Gonçalves, “Free algebras in division rings with an involution”, arXiv:1605.04863 (2016).

Additional references

2 papers in this index state this conjecture (2013–2016). The statement above is taken from the most recent of them; the others are arXiv:1308.6602.

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