Frieze patterns of numbers, introduced in the early 70s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present paper aims to review the original work of Coxeter and the new develo
pments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.
We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hills operators. The space o
f these superfriezes is an algebraic supervariety, which is isomorphic to the space of supersymmetric second order difference equations, called Hills equations.
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair
of frieze patterns and a unimodular $2times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
We consider $G$-graded commutative algebras, where $G$ is an abelian group. Starting from a remarkable example of the classical algebra of quaternions and, more generally, an arbitrary Clifford algebra, we develop a general viewpoint on the subject.
We then give a recent classification result and formulate an open problem.
We study the notion of $Gamma$-graded commutative algebra for an arbitrary abelian group $Gamma$. The main examples are the Clifford algebras already treated by Albuquerque and Majid. We prove that the Clifford algebras are the only simple finite-dim
ensional associative graded commutative algebras over $mathbb{R}$ or $mathbb{C}$. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.
We show that the classical algebra of quaternions is a commutative $Z_2timesZ_2timesZ_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.
