The set of morphisms mapping any Sturmian sequence to a forms together with composition the so-called monoid Sturm. For this monoid, we defne faithful representation by $(3\times 3)$-matrices integer entries. We find three convex cones in $\mathbb{R}^3$ and show that matrix $R \in Sl(\mathbb{Z},3)$ is representing morphism if are invariant under multiplication $R$ or $R^{-1}$. This property off...

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group Bn , which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of the n-punctured sphere by using the cl...

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group Bn , which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of the n-punctured sphere by using the cl...

The Burau representation is a natural action of the braid group Bn on the free Z[t; t−1]{module of rank n−1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n = 3. Moody has shown that it is not faithful for n 9 and Long and Paton improved on Moody’s techniques to bring this down to n 6. Their construction uses...

The notion of conformal algebra appears as an algebraical language describing singular part of operator product expansion (OPE) in conformal field theory [BPZ]. Explicit algebraical exposition of this theory leads to the notion of vertex (chiral) algebra (see, e.g., [B], [DL], [FLM]). Roughly speaking, conformal algebras correspond to vertex algebras by the same way as Lie algebras correspond t...