Search for: All records

We construct finite $$R$$-matrices for the first fundamental representation $$V$$ of two-parameter quantum groups $$U_{r,s}(\mathfrak{g})$$ for classical $$\mathfrak{g}$$, both through the decomposition of $$V\otimes V$$ into irreducibles $$U_{r,s}(\mathfrak{g})$$-submodules as well as by evaluating the universal $$R$$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $$U^{\pm}_{r,s}(\mathfrak{g})$$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $$R$$-matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $$U_{r,s}(\widehat{\mathfrak{g}})$$ for classical $$\mathfrak{g}$$. The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras. 

We generalize the study of standard Lyndon loop words from [16] to a more general class of orders on the underlying alphabet, as suggested in [16, Remark 3.15]. The main new ingredient is the exponent-tightness of these words, which also allows to generalize the construction of PBW bases of the untwisted quantum loop algebra $$U_{q}(L{{\mathfrak{g}}})$$ via the combinatorics of loop words. 

Abstract We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the positive subalgebras of quantum loop algebras of type $$B_{n}$$ and $$G_{2}$$, as well as their Lusztig and RTT (for type $$B_{n}$$ only) integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these $${\mathbb {Q}}(v)$$-algebras (proved earlier in [26] by completely different tools) and generalize the latter to the above $${{\mathbb {Z}}}[v,v^{-1}]$$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $$B_{n}$$ and $$G_{2}$$ Yangians and their Drinfeld-Gavarini duals. All of this generalizes the type $$A_{n}$$ results of [30].