Explore the intricate world of coalgebra symmetry and Discrete Time Invariant Systems (DTIS) in this comprehensive lecture. Delve into key concepts such as primitive product, synthetic representation, and materialization. Examine the role of tensor products in N-dimensional spaces and their application to Hamiltonian systems. Gain insights into the mathematical foundations of discrete time systems and their symmetrical properties. Enhance your understanding of advanced mathematical concepts and their practical applications in physics and engineering.