We present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.

We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.

The ‘polaroid’ property transfers from Banach algebra elements to their tensor product, and hence also to their induced multiplications on ‘ultraprime’ Banach bimodules.

Walter Wilson Stothers was born in Glasgow on 8 November 1946. A third (youngest) son, he had the identical name to his father. From childhood, however, he had always been known by his middle name ‘Wilson’, so that his father, a Glasgow GP, would never be referred to as ‘Old Walter’. His mother, as Jean Young Kyle, had herself graduated in Mathematics in 1927, a rare achievement for a woman at that time. After attending the local primary school 1952–1956, Wilson completed his primary education in the preparatory classes in Allan Glen's School, then a distinguished Glasgow boys school with a scientific emphasis, progressing to the secondary school in 1958 and ending by becoming Dux in 1964. He also played in the school rugby first XV. From 1964–1968 he was a student in the Science Faculty of Glasgow University. His original intention was to take Honours in Chemistry and, indeed, he won the Chemistry prize in his first year. But he excelled in all subjects, winning the Faraday medal in the Intermediate Honours (second year) class in Natural Philosophy (Physics). After that he concentrated on Mathematics and became the top student, gaining a First Class Honours degree, as well as the Cunninghame Medal and a Jack Scholarship to Peterhouse College, Cambridge (1968). Before commencing postgraduate studies he married Andrea Watson in September 1968. At Cambridge from 1968–1971 he studied for a Ph.D. in number theory under the supervision of Peter Swinnerton-Dyer and graduated in 1972 with the thesis Some Discrete Triangle Groups. By then he was becoming aware of the strange realm inhabited by mathematicians that he seemed to be entering. So when his Cambridge room-mate Bob Odoni, at a research meeting they were attending as postgrads, asked, ‘Wilson, do you realise that we are the only normal people here’, Wilson felt compelled to respond, ‘What makes you think that we are normal?’