Neural Networks: According to the Principles of Grassmann Algebra

In this paper, we explore the algebra of quantum idempotents and the quantization of fermions which gives rise to a Hilbert space equal to the Grassmann algebra associated with the Lie algebra. Since idempotents carry representations of the algebra under consideration, they form algebraic varieties and smooth manifolds in the natural topology. In addition to the motivation of linking up mathematical physics with machine learning, it is also shown that by using idempotents and invariant subspace of the corresponding algebras, these representations encode and perhaps provide a probabilistic interpretation of reasoning and relational paths in geometrical terms.