Sobolev Institute of Mathematics

On linear equations over split-octonions

Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations (ax)b=c and a(bx)=c over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial equations in the split-octonions. Moreover, w...

Generating the symmetric group by three prefix reversals

The cubic pancake graphs are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate $\mathrm{Sym}_n$. As the largest prefix reversal of len...

Rota–Baxter Operators on Compact Simple Lie Groups and Algebras

A smooth mapping $$ B \colon G \to G $$ on a Lie group $$ G $$ is called a Rota–Baxter operator if $$ B(g)B(h)=B(gB(g)hB(g)^{-1}) $$ for all $$ g, h \in G$$ . This concept was introduced in 2021 by Guo, Lang, and Sheng as a group analog of weight 1 Rota–Baxter operators on Lie al...

Describing edges in triangle-free 3-polytopes

An edge e in a 3-polytope is of type ( k 1 , k 2 , k 3 , k 4 ) if the set of degrees of the vertices and faces incident with e is majorized by the vector ( k 1 , k 2 , k 3 , k 4 ) . In 1940, Lebesgue proved that every 3-polytope has an edge of one of the types ( 3 , 3 , 3 , ∞ ) ,...