Koszulity of finitely semi-graded algebras

TL;DR

This paper introduces finitely semi-graded algebras, extending connected graded algebras, and investigates their Koszul properties through ideal lattice distributivity, providing new insights into their algebraic structure and series computations.

Contribution

It defines finitely semi-graded algebras, explores their Koszulity via lattice distributivity, and computes their Hilbert and Poincaré series for important examples.

Findings

Finitely semi-graded algebras include key non-ℕ-graded examples from physics.

Koszulity is established for these examples.

Explicit Hilbert and Poincaré series are computed.

Abstract

In this paper, we introduce the class of finitely semi-graded algebras which extends the connected graded algebras finitely generated in degree one. The Koszul behavior of finitely semi-graded algebras is investigated by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincar\'e series and the Yoneda algebra are defined for this class of algebras. Finitely semi-graded algebras include many important examples of non $\mathbb{N}$-graded algebras finitely generated in degree one coming from mathematical physics, and for these concrete examples the Koszulity will be established, as well as, the explicit computation of its Hilbert and Poincar\'e series.