Quantum affine algebra

In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang-Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized.

See also

• Yangian
• Quantum KZ equations
• Littelmann path model

References

• Drinfeld, V. G. (1985), "Hopf algebras and the quantum Yang-Baxter equation", Doklady Akademii Nauk SSSR, 283 (5): 1060–1064, ISSN 0002-3264, MR 0802128
• Drinfeld, V. G. (1987), "A new realization of Yangians and of quantum affine algebras", Doklady Akademii Nauk SSSR, 296 (1): 13–17, ISSN 0002-3264, MR 0914215
• Frenkel, Igor B.; Reshetikhin, N. Yu. (1992), "Quantum affine algebras and holonomic difference equations", Communications in Mathematical Physics, 146 (1): 1–60, Bibcode:1992CMaPh.146....1F, doi:10.1007/BF02099206, ISSN 0010-3616, MR 1163666
• Jimbo, Michio (1985), "A q-difference analogue of U(g) and the Yang-Baxter equation", Letters in Mathematical Physics, 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588, ISSN 0377-9017, MR 0797001
• Jimbo, Michio; Miwa, Tetsuji (1995), Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC, ISBN 978-0-8218-0320-2, MR 1308712