Algebraic geometry over division rings

Abstract: We shall survey recent developments concerned with foundational aspects of algebraic geometry over division rings:

1. A quaternionic Nullstellensatz for the ring R of polynomials in n central variables over the quaternion algebra H, in both abstract form (due to the author and Alon) and explicit form (due to M. Aryapoor).

2. A theorem about the geometry of zero sets of polynomials in R: If a polynomial vanishes on all common zeros with commuting coordinates of a left ideal J in R , then it vanishes on all common zeros of J in H^n. This result confirmed a conjecture of Gori, Sarfatti and Vlacci.

3. Study of contraction properties of one-sided ideals in polynomial rings over division rings. In particular, we show that if M is a maximal left ideal in the polynomial ring D[x], where D is a division ring, then the contraction of M to D need not be maximal. This resolved a question of Amitsur and Small from 1978. We shall discuss connections between this question to recent works and the implications to algebraic geometric over division rings.

Joint works with Gil Alon, Adam Chapman.

quantum algebrarings and algebras

Audience: researchers in the topic

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European Non-Associative Algebra Seminar

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