Matrix Genetics and Algebraic Properties of the Multi-Level System of Genetic Alphabets

The article is devoted to algebraic properties of the multi-level system of molecular-genetic alphabets. It leads to help solve the problem of algebraic unity of inherited information systems in living matter. These algebraic properties are revealed by means of Kronecker families of matrix forms of a presentation of molecular-genetic alphabets. A family of genetic (8x8)-matrices shows unexpected connections of the genetic system with Rademacher and Walsh functions and with special Hadamard matrices which are well-known in theory of noise-immunity coding and digital communication. Decompositions of such genetic (8x8)-matrices on the basis of the known principle of dyadic-shifts lead to sets of 8 sparse matrices. Each of these sets is closed in relation to multiplication and defines a special algebra of 8-dimensional hypercomplex numbers. Mathematical aspects of these 8-dimensional algebras are presented in connection with metric vector spaces, the sequency theory by Harmuth and some methods of spectral analysis. The diversity of known dialects of the genetic code can be analyzed from the viewpoint of these algebras. Our results are discusse