Kalgebras

Kalgebras are a class of algebraic structures studied in abstract algebra. Formally, a Kalgebra over a field F is a pair (A, κ) where A is an associative (and typically unital) algebra over F, and κ: A → F is a linear functional, called the Kal form, that satisfies κ(ab) = κ(ba) for all a, b in A. The Kal form need not be nondegenerate, but when it is, the algebra is said to be nondegenerate. The defining symmetry implies that κ vanishes on all commutators, since κ([a, b]) = κ(ab − ba) = 0 for all a, b.

The Kal form provides a trace-like invariant on A and induces a natural pairing between A and its dual via a ↦ κ(a, −). If κ is nondegenerate, this pairing yields a duality between A and its dual space, giving structural information about A and its modules. Finite-dimensional Kalgebras over F are particularly amenable to study via representation-theoretic methods that use κ to analyze traces and characters of representations.

Examples and constructions: the full matrix algebra Mn(F) with κ(A) = trace(A) is a Kalgebra, since trace(AB) = trace(BA). Finite direct sums of matrix algebras carry Kal forms given by sums of traces on each block. Subalgebras closed under κ inherit a Kal form by restriction. The tensor product of Kalgebras (A, κA) and (B, κB) can be given a Kal form by (κA ⊗ κB)(a ⊗ b) = κA(a) κB(b).

Relations and scope: Kalgebras are related to trace forms and, in special cases, connect to Frobenius-type dualities when the Kal form is nondegenerate and compatible with the multiplication. They provide a flexible framework for studying bilinear invariants on algebras and their module categories.