Quantum and Braided Linear Algebra

الأصل · EN

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. We develop this interpretation 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. We show 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).