We consider the R-matrix of quantum toroidal algebra type gl1, both abstractly and in Fock space representations. provide a survey certain point view on this object which involves elliptic Hall shuffle algebras, we show how to obtain explicit formulas.

We describe a quantum Lie algebra based on the Cremmer-Gervais R-matrix. The algebra arises upon a restriction of an infinite-dimensional quantum Lie algebra.

• Euclid synthetic geometry 300 BC • Descartes analytic geometry 1637 • Gauss – complex algebra 1798 • Hamilton – quaternions 1843 • Grassmann – Grasmann Algebra 1844 • Cayley – Matrix Algebra 1854 • Clifford – Clifford algebra 1878 • Gibbs – vector calculus 1881 – used today • Sylvester – determinants 1878 • Ricci – tensor calculus 1890 • Cartan – differential forms 1908 • Dirac, Pauli – spin ...