Quantum and braided linear algebra

Abstract

Quantum matrices A(R) are known for every R matrix obeying the quantum Yang–Baxter equations. It is also known that these act on ‘‘vectors’’ given by the corresponding Zamalodchikov algebra. This interpretation is developed in detail, distinguishing between two forms of this algebra, V(R) (vectors) and V*(R) (covectors). A(R)→V(R21)⊗V*(R) is an algebra homomorphism (i.e., quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. It is shown that if V(R) and V*(R) are endowed with the necessary braid statistics Ψ then their braided tensor-product V(R)⊗_V*(R) is a realization of the braided matrices B(R) introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from B(R) act on themselves by conjugation in a way impossible for the quantum groups obtained from A(R).