The Idris Algebra: Structure, Representations, Growth, and Homological Dimension

The Idris algebra is introduced as a new class of associative algebras equipped with a finite-dimensional collapse ideal and a faithful expansion action. The paper develops the structural theory of these algebras, establishing a semidirect product decomposition, describing the collapse ideal, and analyzing the commutator action of the expansion algebra. Representation theory is studied both for the collapse ideal and for the full algebra, including a classification of finite-dimensional modules over the collapse ideal. The center, derivations, and automorphisms are computed, and a natural filtration is introduced, leading to a description of the associated graded algebra. The Gelfand–Kirillov dimension and global homological dimension of an Idris algebra are shown to coincide with those of its expansion algebra. Several examples are provided, illustrating how the theory applies to polynomial algebras and universal enveloping algebras. The work establishes a foundation for further study of Idris algebras within noncommutative algebra and representation theory.