Jordanian solutions of simplex equations (Q1803243)

The quantum Yang-Baxter, or triangle equation can be included as a first member in the sequence of simplex equations, which are called Frenkel- Moore \(N\)-simplex. This is an equation for an operator in the \(N\)th power \(V^{\times N}\) of a space \(V\), i.e. for a tensor \(R^{\widehat\alpha_ 1\cdots \alpha_ i\cdots \alpha_{N+1}}_{\beta_ 1\cdots \widehat\beta_ i\cdots \beta_{N+1}}\delta^{\alpha_ i}_{\beta_ i}\), where the hat over an index means that this index is omitted, then the quantum \(N\)-simplex equation has the form \(R_ 1 R_ 2\cdots R_{N+1}= R_{N+1} R_ N\cdots R_ 1\) (for \(N=2\), one obtains the Yang-Baxter equation). In this paper the authors investigate an \(N\)-simplex generalization of the Jordanian \(R\)-matrix (which gives rise to the Jordanian quantum group \(SL^ J(2)\)). The authors prove that the classical Jordanian \(R\)-matrix can be quantized for any \(N\), giving rise to nontrivial solutions of the quantum \(N\)-simplex equations which are the exponents of the classical ones.