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Perfect crystals and \(q\)-deformed Fock spaces. (English) Zbl 0959.17014
Summary: In [E. Stern, Int. Math. Res. Not. 4, 201-220 (1995; Zbl 0823.17042) and M. Kashiwara, T. Miwa and E. Stern, Sel. Math. 1, 787-805 (1995; Zbl 0857.17013)] the semi-infinite wedge construction of level 1 \(U_q(A_n^{(1)})\) Fock spaces and their decomposition into the tensor product of an irreducible \(U_q(A_n^{(1)})\)-module and a bosonic Fock space are given. Here a general scheme for the wedge construction of \(q\)-deformed Fock spaces using the theory of perfect crystals is presented.
Let \(U_q({\mathfrak g})\) be a quantum affine algebra. Let \(V\) be a finite-dimensional \(U_q'({\mathfrak g})\)-module with a perfect crystal base of level \(l\). Let \(V_{\text{aff}} \simeq V\otimes \mathbb{C} [z,z^{-1}]\) be the affinization of \(V\), with crystal base \((L_{\text{aff}}, B_{\text{aff}})\). The wedge space \(V_{\text{aff}} \wedge V_{\text{aff}}\) is defined as the quotient of \(V_{\text{aff}} \otimes V_{\text{aff}}\) by the subspace generated by the action of \(U_q ({\mathfrak g}) [z^a\otimes z^b+ z^b \otimes z^a]_{a,b\in \mathbb{Z}}\) on \(v\otimes v\) (\(v\) an extremal vector). The wedge space \(\bigwedge^r V_{\text{aff}}\) \((r\in \mathbb{N})\) is defined similarly. Normally ordered wedges are defined by using the energy function \(H: B_{\text{aff}} \otimes B_{\text{aff}}\to \mathbb{Z}\). Under certain assumptions, it is proved that normally ordered wedges form a base of \(\bigwedge^r V_{\text{aff}}\).
A \(q\)-deformed Fock space space is defined as the inductive limit of \(\bigwedge^r V_{\text{aff}}\) as \(r\to \infty\), taking along the semi-infinite wedge associated to a ground state sequence. It is proved that normally ordered wedges form a base of the Fock space and that of Fock space has the structure of an integrable \(U_q({\mathfrak g})\)-module. An action of the bosons, which commute with the \(U_q'({\mathfrak g})\)-action, is given on the Fock space. It induces the decomposition of the \(q\)-deformed Fock space into the tensor product of an irreducible \(U_q({\mathfrak g})\)-module and a bosonic Fock space.
As examples, Fock spaces for types \(A_{2n}^{(2)}\), \(B_n^{(1)}\), \(A_{2n-1}^{(2)}\), \(D_n^{(1)}\) and \(D_{n+1}^{(2)}\) at level 1 and \(A_1^{(1)}\) at level \(k\) are constructed. The commutation relations of the bosons in each of these cases are calculated, using two point functions of vertex operators.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B69 Vertex operators; vertex operator algebras and related structures
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[1] V. Chari and A. Pressley.A Guide to Quantum Groups. Cambridge University Press, 1994. · Zbl 0839.17009
[2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa. Transformation Groups for soliton equations – Euclidean Lie algebras and reduction of the KP hierarchy.Publ. R.I.M.S., Kyoto Univ. 18 (1982), 1077–1110. · Zbl 0571.35103 · doi:10.2977/prims/1195183297
[3] E. Date, M. Jimbo and M. Okado. Crystal bases andq-vertex operators.Commun. Math. Phys. 155 (1993), 47–69. · Zbl 0790.17007 · doi:10.1007/BF02100049
[4] E. Date and M. Okado. Calculation of excitation spectra of the spin model related with the vector representations of the quantized affine algebra of typeA n (1) .Int. J. Mod. Phys. A 9 (1994), no. 3, 399–417. · Zbl 0986.82500 · doi:10.1142/S0217751X94000194
[5] T. Hayashi.q analogues of Clifford and Weyl algebras – spinor and oscillator representations of quantum enveloping algebras.Commun. Math. Phys. 127 (1990), 129–144. · Zbl 0701.17008 · doi:10.1007/BF02096497
[6] M. Idzumi, K. Iohara, M. Jimbo, T. Miwa, T. Nakashima and T. Tokihiro. Quantum affine symmetry in vertex models.Int. J. Mod. Phys. A 8 (1993), 1479–1511; (hepth/9208066). · doi:10.1142/S0217751X9300062X
[7] M. Jimbo. QuantumR-matrix for the generalised Toda system.Commun. Math. Phys. 102 (1986), 537–547. · Zbl 0604.58013 · doi:10.1007/BF01221646
[8] M. Jimbo and T. Miwa. Algebraic analysis of solvable lattice models. CBMS Regional conference series in mathematics, vol. 85, A.M.S., Providence, Rhode Island, 1995. · Zbl 0828.17018
[9] P. D. Jarvis and C. M. Yung. Combinatorial description of the Fock representation of the affine Lie algebrago(.Lett. Math. Phys. 30 (1994), 45–52. · Zbl 0787.05092 · doi:10.1007/BF00761421
[10] M. Kashiwara. Crystallising theq-analogue of universal enveloping algebras.Commun. Math. Phys. 133 (1990), 249–260. · Zbl 0724.17009 · doi:10.1007/BF02097367
[11] M. Kashiwara. Global crystal bases of quantum groups.Duke Math. J. 69 (1993), 455–485. · Zbl 0774.17018 · doi:10.1215/S0012-7094-93-06920-7
[12] V. Kac, D. A. Kazhdan, J. Lepowsky and R. L. Wilson. Realization of the basic representations of the Euclidean Lie algebras.Adv. Math. 42 (1981), 83–112. · Zbl 0476.17003 · doi:10.1016/0001-8708(81)90053-0
[13] S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki. Affine crystals and vertex models.Int. J. Mod. Phys. A 7, Suppl. 1A (1992), 449–484; Proceedings of the RIMS Project 1991 ”Infinite Analysis”. · Zbl 0925.17005 · doi:10.1142/S0217751X92003896
[14] S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki. Perfect crystals of quantum affine Lie algebras.Duke Math. J. 68 (1992), 499–607. · Zbl 0774.17017 · doi:10.1215/S0012-7094-92-06821-9
[15] M. Kashiwara, T. Miwa and E. Stern. Decomposition ofq-deformed Fock spaces.Selecta Math. 1 (1995), no. 4, 787–805; (q-alg/9508006). · Zbl 0857.17013 · doi:10.1007/BF01587910
[16] J. Lepowsky and R. L. Wilson. Construction of the affine Lie algebraA 1 (1) .Commun. Math. Phys. 62 (1978), 43–53. · Zbl 0388.17006 · doi:10.1007/BF01940329
[17] K. C. Misra and T. Miwa. Crystal base for the basic representation ofU q (ŝl(n)).Commun. Math. Phys. 134 (1990), 79–88. · Zbl 0724.17010 · doi:10.1007/BF02102090
[18] E. Stern. Semi-infinite wedges and vertex operators.Internat. Math. Res. Notices 4 (1995), 201–220; (q-alg/9505030). · Zbl 0823.17042 · doi:10.1155/S107379289500016X
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