30-year-old mathematical hypothesis disproved
A research collaboration between BTH, University West and the University of Santander in Colombia has led to a new discovery in mathematics. The discovery concerns separable algebras and their gradations.
“It is a great feeling to be able to produce counter-examples, and thereby prove something that three of the world’s foremost algebra researchers did not believe was possible”, says Johan Öinert at BTH.
The discovery concerns the hypothesis that all separable graded algebras must be strongly graded, as formulated 30 years ago by the three mathematicians Lieven Le Bruyn, Michel Van den Bergh and Freddy Van Oystaeyen. Now, in a recently published article, this hypothesis has been disproved in a research collaboration between Professor Johan Öinert at BTH, Professor Patrik Nystedt at University West and Professor Héctor Pinedo at the University of Santander in Colombia.
“We have introduced a new type of gradation which does not need to be strong, but which is very closely related to strong gradations. We call these ‘epsilon-strong gradations’. In addition, we have provided a precise description of when algebras with epsilon-strong gradations are separable. This is an extension of previous results for strongly graded algebras”, Johan explains.
“Finally, we have proven that a graded algebra which is separable does not necessarily come from a strong gradation. It is possible to find examples of epsilon-strong graded algebras that are separable. It is this discovery that proves the falsity of the Belgians’ assumption. In fact, we find not only one example, which would have been entirely sufficient, but an endless number of examples”, he says.
The theory of epsilon-strong graded algebras is now being further developed by many different researchers. For example, doctoral student Daniel Lännström at BTH is basing his doctoral thesis work on the discoveries in the article.
Contact Johan Öinert via email: johan.oinert@bth.se
11 December 2018