Semi-symmetric algebras: General Constructions. Part II

TL;DR

This paper extends the theory of semi-symmetric algebras by exploring their duals, bilinear forms, coalgebra structures, and inner products, unifying the treatment of tensor, symmetric, and exterior algebras.

Contribution

It provides a comprehensive and unified analysis of semi-symmetric algebras, including their duals and structural properties, building on previous constructions and classifications.

Findings

Unified treatment of semi-symmetric, tensor, symmetric, and exterior algebras.

Analysis of duals and bilinear forms of semi-symmetric algebras.

Structural insights into coalgebra and inner product properties.

Abstract

In our paper Semi-symmetric Algebras: General Constructions, J. Algebra, 148 (1992), pp. 479-496, we present the construction of the semi-symmetric algebra of a module over a commutative ring with unit, which generalizes the tensor algebra, the symmetric algebra, and the exterior algebra, deduce some of its functorial properties, and prove a classification theorem. In the present paper we continue the study of the semi-symmetric algebra and discuss its graded dual, the corresponding canonical bilinear form, its coalgebra structure, as well as left and right inner products. Here we present a unified treatment of these topics whose exposition in N. Bourbaki, Alg\` ebre, Chapitres 1--3, Hermann, Paris 1970, is made simultaneously for the above three particular (and, without a shadow of doubt - most important) cases.