Abstract
This Letter concerns an extension of the quantum spinor construction of \({U}_q (\widehat{\mathfrak{g}\mathfrak{l}}(n))\). We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, and construct quantum spinor representations of \({\text{U}}_q (\hat {\mathfrak{o}}(N))\).
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References
Bernard, D.: Comm. Math. Phys. 165(1989), 555–568.
Date, E. and Okado, M.: Calculation of excitation spectra of the spin model related with the vector representation of the quantized affine algebra of type \(A_n^{(1)} \), Internat. J. Modern Phys. A. 9(3) (1994).
Davies, E. and Okado, M.: Excitation spectra of spin models constructed fromquantization affine algebras of type B (1), \(D_n^{(1)} \), hep-th/9506201.
Davis, B., Foda, O., Jimbo, M., Miwa T. and Nakayashiki, A.: Diagonalization of the XXZ Hamiltonian by vertex operators, Comm. Math. Phys. 151(1993), 89–153.
Ding, J.: Spinor representations of Uq(\(\widehat{\mathfrak{g}\mathfrak{l}}\)(n) and quantum Boson–Fermion correspondence RIMS-1043, q-alg/9510014.
Ding, J., Frenkel, I.B.: Isomorphism of two realizations of quantum affine algebra Uq(\(\widehat{\mathfrak{g}\mathfrak{l}}\)(n), Comm. Math. Phys. 156(1993), 277–300.
Ding, J. and Frenkel, I. B.: Spinor and oscillator representations of quantum groups, in: Lie Theory and Geometry in Honor of Bertram Kostant, Progr. in Math. 123, Birkhauser, Boston, 1994.
Drinfeld, V. G.: Hopf algebra and the quantum Yang–Baxter Equation, Dokl. Akad. Nauk. SSSR 283(1985), 1060–1064.
Drinfeld, V. G.: Quantum Groups, ICM Proceedings, New York, Berkeley, 1986, 798–820.
Drinfeld, V. G.: New realization of Yangian and quantum affine algebra, Soviet Math. Dokl. 36 (1988), 212–216.
Faddeev, L. D., Reshetikhin, N. Yu and Takhtajan, L. A.: Quantization of Lie groups and Lie Algebras, Yang-Baxter Equation in Integrable Systems, Adv. Ser. Math. Phys. 10, World Scientific, Singapore, 1989, pp. 299–309.
Feingold, A. J. and Frenkel, I. B.: Classical affine Lie algebras, Adv. Math. 56(1985), 117–172.
Foda, O., Iohara, H., Jimbo, M., Kedem, R., Miwa, T. and Yan, H.: Notes on highest weight modules of the elliptic algebra \(\mathfrak{A}_{p,q} (\widehat{\mathfrak{s}\mathfrak{l}}_2 )\), To appear in T. Inami and R. Sasahi (eds), Quantum Field Theory, Integrable Models and Beyond, Supplements of Progr. Theoret. Phys.
Frenkel, I. B.: Spinor representation of affine Lie algebras, Proc. Natl. Acad. Sci. USA 77(1980) 6303–6306.
Frenkel, I. B.: Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory, J. Funct. Anal. 44(1981), 259–327.
Frenkel, I. B., Lepowsky, J. and Meurman, A.: Vertex Operator Algebras and the Monster, Academic Press, Boston, 1988.
Frenkel, I. B. and Kac, V. G.: Basic representations of affine Lie algebras and dual resonance model, Invent. Math. 62(1980), 23–66.
Frenkel, I. B. and Jing, N.: Vertex representations of quantum affine algebras, Proc. Natl. Acad. Sci. USA 85(1988), 9373–9377.
Frenkel, I. B. and Reshetikhin, N. Yu.: Quantum affine algebras and holomorphic difference equation, Comm. Math. Phys. 146(1992), 1–60.
Garland, H.: The arithmetic theory of loop groups, Publ. Math. IHES 52(1980) 5–136.
Hayashi, T.: Q-analogue of Clifford and Weyl algebras–spinor and oscillator representation of quantum enveloping algebras, Comm. Math. Phys. 127(1990), 129–144.
Jimbo, M.: A q-difference analogue of \({\text{U}}(\mathfrak{g})\) and Yang–Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.
Jimbo, M.: A q-analogue of U q \(\mathfrak{g}l\)(n+1)), Hecke algebra and the Yang–Baxter equation, Lett. Math. Phys. 11(1986), 247–252.
Jimbo, M.: Quantum R-matrix for the generalized Toda systems, Comm. Math. Phys. 102(1986), 537–548.
Jimbo, M.: Introduction to the Yang–Baxter equation, Internat. J. Modern Phys. A 4(15), (1990), 3758–3777.
Jimbo, M., Miki, K., Miwa, T. and Nakayashiki, A.: Correlation functions of the XXZ model for Δ < _1, Phys. Lett. A 168(1992), 256–263.
Jing, N., Kang, S. and Koyama, Y.: Vertex operators of quantum affine algebra \(D_n^{(1)} \), to appear in Comm. Math. Phys.
Kac, V. G.: Infinite Dimensional Lie Algebras, 3rd edn, Cambridge University Press, Cambridge, 1990.
Kac, V. G. and Peterson, D. H.: Spinor and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA 78(1981) 3308–3312.
Kang, S., Kashiwara, M., Misra, K., Miwa, T. Nakashima, T. and Nakayashiki, A.: Affine crystals and vertex models, Internat. J. Modern Phys. A 7(Supp 1.1A) (1992), 449–484.
Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70(1988), 237–249.
Miki, K.: Creation/annihilation operators and form factors of XXZ model, Phys. Lett. A 186 (1994), 217–224.
Okado, M.: Private communication.
Reshetikhin, N. Yu. and Semenov-Tian-Shansky, M. A.: Central extensions of quantum current groups, Lett. Math. Phys. 19(1990), 133–142.
Smirnov, F. A.: Introduction to quantum groups and integrable massive models of quantum field theory, in: Mo-Lin Ge, Bao-Heng Zhao (eds), Nankai Lectures on Mathematical Physics, World Scientific, Singapore, 1990.
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DING, J. Spinor Representations of \({U}_q (\hat {\mathfrak{o}}(N))\) ). Letters in Mathematical Physics 39, 81–94 (1997). https://doi.org/10.1023/A:1007359702303
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DOI: https://doi.org/10.1023/A:1007359702303