Differential calculus on the inhomogeneous quantum groups \(\text{IGL}_ q(n)\)
The quantum inhomogeneous group \(\text{IGL}_ q(n)\) is realized as a quantum subgroup in \(\text{GL}_ q(n+1)\). The bicovariant differential calculus on \(\text{IGL}_ q(n)\) is obtained via a projection from \(\text{GL}_ q (n+1)\) and its differential calculus. Express the index \(A= (0,1,\dots, n)\) as \(A=(0,a)\) and similarly for \(B\), \(C\), \(D\). Order the indices \(AB\) as \(00\), \(0b\), \(a0\), \(ab\). The basic observation is that the \(R\)-matrix of \(\text{GL}_ q(n+1)\) can be written as \[ R^{AD}_{CD}= \begin{pmatrix} q &0 &0 &0\\ 0 &1 &0 &0\\ 0 &\lambda &1 &0\\ 0 &0 &0 &R^{ab}_{cd}\end{pmatrix}, \qquad \lambda= q-q^{-1}, \] where \(R^{ab}_{cd}\) is the \(R\)-matrix for \(\text{GL}_ q (n)\). The projection (restriction morphism) \(\text{GL}_ q (n+1)\to \text{IGL}_ q(n)\) is defined by \(T^ 0_ a=0\), \(T^ a_ 0= x^ a\), \(T^ 0_ 0=u\). The algebra \(\text{Fun} (\text{IGL}_ q (n))\) is generated by \(x^ a\), \(u\), \(T^ a_ b\) and, in addition, by \(v= u^{-1}\), \(\xi=(\text{deg}_ q T^ a_ b)^{-1}\). The differential calculus is then obtained by letting \(\omega_ 0^ a= \omega_ 0^ 0= 0\), \(\chi^ 0_ a= \chi^ 0_ 0=0\). This way the author rederives his own result from [Phys. Lett. B 298, 335-338 (1993)].
quantum inhomogeneous group \(\text{IGL}_ q(n)\)
scientific article; zbMATH DE number 540166