Quantum and Braided Linear Algebra

Shahn Majid

doi:10.1063/1.530193

paper

Quantum and Braided Linear Algebra

arXiv:hep-th/9208006 · doi:10.1063/1.530193

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. We develop this interpretation in detail, distinguishing between two forms of this algebra, $V (R)$ (vectors) and $V^{*} (R)$ (covectors). $A(R)\to V(R_{21})\tens V^*(R)$ 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 $V(R)\und\tens V^*(R)$ 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 $A (R)$ .

27 pages

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Deformed Traces and Covariant Quantum Algebras for Quantum Groups $G L_{q p} (2)$ and $G L_{q p} (1∣1)$

Quantum and Braided Linear Algebra

Abstract

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