tensorbara

Tensorbara is a term used in fiction and in some speculative mathematical literature to denote a generalized tensor-like structure equipped with an additional involutive bar operation. A tensorbara consists of a graded vector space T = ⨁_{p,q∈Z} T^p_q over a field, together with a family of multilinear products ⊗: T^p_q × T^r_s → T^{p+r}_{q+s}. The bar operation is an anti-linear involution Bar: T^p_q → T^q_p that interchanges covariant and contravariant grading and satisfies Bar(x ⊗ y) = Bar(x) ⊗ Bar(y). The structure is designed to model dualities, contractions, and index transformations in a single algebraic framework.

Key properties include closure under contraction maps that lower the total grading and the existence of a trace form tr: T^p_p → F that is Bar-invariant. Representations of tensorbara appear in toy models of duality in physics and in certain categories that resemble tensor categories with their own bar duality.

Relation to existing concepts: Tensorbara generalizes ordinary tensor algebras by including the bar operation; it shares ideas with graded algebras, duality operations, and tensor categories, but it is not standardized in mainstream mathematics.

Applications and reception: In fiction, tensorbara can be used to illustrate dualities between spaces; in speculative mathematics, it is used as a testbed for theories of categorification and higher algebra. The term remains niche and provisional.

See also: Tensor algebra, graded algebra, dual space, bar involution.