2025/03/11 by Laurie, Duncan
#17B37 #17B67 #20F36 #81R50 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2503.08839
We introduce a new topological coproduct Δψu for quantum toroidal algebras Uq(\mathfrakgtor) in all untwisted types, leading to a well-defined tensor product on the category \widehatOint of integrable representations. This is defined by twisting the Drinfeld coproduct Δu with an anti-involution ψ of Uq(\mathfrakgtor) that swaps its horizontal and vertical quantum affine subalgebras. Other applications of ψ include generalising the celebrated Miki automorphism from type A, and an action of the universal cover of SL2(ℤ). Next, we investigate the ensuing tensor representations of Uq(\mathfrakgtor), and prove quantum toroidal analogues for a series of influential results by Chari-Pressley on the affine level. In particular, there is a compatibility with Drinfeld polynomials, and the product of irreducibles is generically irreducible. We moreover show that the q-character of a tensor product is equal to the product of q-characters for its factors. Furthermore, we obtain R-matrices with spectral parameter which provide solutions to the (trigonometric, quantum) Yang-Baxter equation, and endow \widehatOint with a meromorphic braiding. These moreover give rise to a commuting family of transfer matrices for each module.