A new new coproduct on quantum loop algebras

Quantum loop algebras generalize $U_q(\widehat{\mathfrak{g}})$ for simple Lie algebras $\mathfrak{g}$, and they include examples such as quantum affinizations of Kac-Moody Lie algebras, K-theoretic Hall algebras of quivers, and BPS algebras for toric Calabi-Yau threefolds. In the present paper, we define a coproduct on general quantum loop algebras, which coincides with the Drinfeld-Jimbo coproduct in the particular case of $U_q(\widehat{\mathfrak{g}})$ . We investigate the consequences of our construction for the representation theory of quantum loop algebras, particularly for tensor products of modules and R-matrices.

💡 Research Summary

The paper introduces a novel “new‑new” coproduct on the broad class of quantum loop algebras, extending the familiar Drinfeld–Jimbo coproduct beyond the traditional quantum affine algebras. Starting from a finite index set I, a characteristic‑zero field K, and a family of rational functions ζ_{ij}(x) satisfying mild growth conditions, the author constructs the quantum loop algebra U generated by e_{i,d}, f_{i,d} and Cartan currents ϕ^{±}_{i,d}. The pre‑quantum loop algebras eU^{±} are presented via generating series e_i(z), f_i(z) with the exchange relations (28)–(29).

A key technical device is the shuffle algebra realization: the author defines vector spaces V^{±} of color‑symmetric Laurent polynomials equipped with a shuffle product (31). Homomorphisms Υ^{±}: eU^{±} → V^{±} embed the pre‑quantum algebras into shuffle subalgebras S^{±}=Im Υ^{±}. This representation makes the otherwise abstract relations concrete: the “wheel conditions” on the polynomials encode the ζ_{ij} exchange factors.

To obtain a bialgebra structure, the Cartan currents are adjoined, yielding extended algebras U^{≥0}=U^{+}⊗K