Before revision:  Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for grid cells' path integration based on two coupled matrix Lie algebras that underlie two coupled rotation systems that mirror the agent's self-motion : (1) When the agent moves along a certain direction, the vectoris rotated by a generator matrix.
After revision:  We then study a minimally simple prototype model where the local change is a linear transformation of the vector. This linear model gives rise to explicit algebraic structure in terms of matrix Lie algebra and matrix Lie group, as well as explicit geometric structure where the self-motion : (1) When the agent moves along a certain direction, the vectoris rotated by a generator matrix.