On Pentagon, Ten-Term, and Tetrahedron Relations

Abstract

It is shown that the tetrahedron equation under the substitution , where P 23 is the permutation operator, is reduced to a pair of pentagon and one ten-term equations on operators S and . Examples of infinite dimensional solutions are found. O-doubles of Novikov, which generalize the Heisenberg double of a Hopf algebra, provide a particular algebraic solution to the problem.