Kaehler Algebra: Idempotents for Solutions with Symmetry in Metric Spaces

TL;DR

This paper explores the use of Kaehler algebra and its tensor product with tangent Clifford algebra to address symmetry solutions in metric spaces, overcoming limitations imposed by non-commutativity and expanding the algebraic framework.

Contribution

It introduces a commutative subalgebra within Kaehler algebra that removes restrictions on the number of idempotents, enabling more complex symmetry solutions in differential geometry.

Findings

The tensor product with tangent Clifford algebra allows more idempotents in solutions.

Mirror elements form a commutative subalgebra simplifying symmetry representations.

Total angular momentum remains linear but not vectorial in Kaehler calculus.

Abstract

With his Clifford algebra of differential forms, Kaehler's algebra addresses the overlooked manifestation of symmetry in the solutions of exterior systems. In this algebra, solutions with a given symmetry are members of left ideals generated by corresponding idempotents. These combine with phase factors to take care of all the dependence of the solution on $x^{i}$ and $% dx^{i}$ for each $\partial /\partial x^{i}=0$ symmetry. The maximum number of idempotents (each for a one-parameter group) that can, therefore, go into a solution is the dimensionality $n$ of the space. This number is further limited by non-commutativity of idempotents. We consider the tensor product of Kaehler algebra with tangent Clifford algebra, i.e. of valuedness. It has a commutative subalgebra of "mirror elements", i.e. where the valuedness is dual to the differential form (like in $dx\mathbf{i}$, but not in $dx\mathbf{j}$). It removes the aforementioned limitation in the number of idempotent factors that can simultaneously go into a solution. In the Kaehler calculus, which is based on the Kaehler algebra, total angular momentum is linear in its components, even it is not valued in tangent algebra. In particular, it is not a vector operator. We carry Kaehler's treatment over to the aforementioned commutative algebra. Non-commutativity thus ceases to be a limiting factor in the formation of products of idempotents representing each a one-parameter symmetry. The interplay of factors that emerges in the expanded set of such products is inimical to operator theory. For practical reasons, we stop at products involving three one-parameter idempotents ---henceforth call ternary--- even though the dimensionality of the all important spacetime manifold allows for four of them.