Twisted quantum affine algebras (Q1265738)

In two earlier papers [Commun. Math. Phys. 142, 261-283 (1991; Zbl 0739.17004) and Can. Math. Soc. Conf. Proc. 16, 59-78 (1995; Zbl 0855.17009)] the authors classified the finite-dimensional irreducible representations of untwisted quantum affine algebras by a one-to-one correspondence of the highest weights with \(n\)-tuples of polynomials, \(n\) being the rank of the underlying finite-dimensional Lie algebra. Now these results are extended to the more complicated case of twisted quantum affine algebras. Once more the finite-dimensional irreducible representations may be parametrized by \(n\)-tuples of polynomials, but now \(n\) denotes the rank of the fixed point subalgebra of the diagram automorphism and the correspondence between highest weights and \(n\)-tuples of polynomials is more complicated than in the untwisted case. Untwisted quantum affine algebras can be reduced in some sense to the quantum affine \(sl_2\), in the twisted case \(U_q(L(sl_3)^\tau)\) is needed in addition. \(\tau\) denotes the unique nontrivial diagram automorphism of \(sl_3(\mathbb{C})\), \(U_q(L(sl_3)^\tau)\) is the quotient of the Hopf algebra \(U_q((sl_3)^\tau)\) by the ideal generated by \(c-1\), and \(c\) denotes a certain central, group-like element. Hence the paper is centrally concentrated on the investigation of \(U_q(L(sl_3)^\tau)\), its subalgebras and representations. Some details of proofs are omitted claiming similarity to proofs in the untwisted case.