The n-ary algebra of tensors and of cubic and hypercubic matrices

TL;DR

This paper introduces a new family of totally associative (2k+1)-ary products on tensor and matrix spaces, generalizing cubic and hypercubic matrices, and explores their algebraic properties and operads.

Contribution

It defines and analyzes a novel class of totally associative (2k+1)-ary products on tensors and matrices, extending the algebraic framework of cubic and hypercubic matrices.

Findings

Defined a (2k+1)-ary product on tensor spaces with s_k-totally associativity.

Extended the product to cubic and hypercubic matrices.

Computed the quadratic operads and their duals for these algebraic structures.

Abstract

We define a ternary product and more generally a (2k+1)-ary product on the vector space T^p_q(E) of tensors of type (p, q) that is contravariant of order p, covariant of order q and total order (p+q). This product is totally associative up to a permutation s_k of order k (we call this property a s_k-totally associativity). When p=2 and q=1, we obtain a (2k+1)-ary product on the space of bilinear maps on E with values on E, which is identified to the cubic matrices. Then we obtain a (2k+1)-ary product on the space of cubic matrices. If we call a l-matrix a square tableau with lx...xl entrances (if l=3 we have the cubic matrices and we speak about hypercubic matrices as soon as l >3), then the (2k+1)-ary product on T^p_q(E) gives a (2k+1)-product on the space of (p+q)-matrices. We describe also all these products which are s_k-totally associative. We compute the corresponding quadratic operads and their dual.