Multiary gradings

TL;DR

The paper extends graded algebra theory to polyadic structures by introducing multiary $\mathsf{G}$-graded polyadic algebras, where both algebraic and grading-group arities are free to vary and are later constrained by quantization rules. It derives key results such as $|G|=\ell_m(m-1)+1$ for general strong gradings and $n'=n$ in the strongly graded case, as well as $\ell_{n'}(n'-1)=\ell_n(n-1)$ for higher power gradings, and classifies graded homomorphisms via pairs $(\Phi,\Psi)$. The authors provide concrete polyadic examples including derived and strictly nonderived ternary superalgebras, polynomial algebras over $n$-ary matrices graded by polyadic integers, and higher-power gradings that exhibit inherently polyadic phenomena. This framework reveals new structural constraints and opens avenues for future work in representation theory, cohomology, physics applications, and geometric interpretations of polyadic graded systems, with potential to impact areas such as Nambu mechanics and generalized supersymmetry.

Abstract

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, and concrete examples including ternary superalgebras and polynomial algebras over $n$-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility.

Theorems & Definitions (21)

• Remark 3.3
• Definition 3.4
• Definition 3.5
• proof
• Proposition 3.7
• proof
• Remark 3.8
• Remark 3.9
• Theorem 3.10
• proof