Perfect crystals andq-deformed Fock spaces |
| |
Authors: | M Kashiwara T Miwa J -U H Petersen C M Yung |
| |
Institution: | (1) Research Institute for Mathematical Sciences, Kyoto University, 606-01 Kyoto, Japan |
| |
Abstract: | In S], KMS] the semi-infinite wedge construction of level 1U
q
(A
n
(1)
) Fock spaces and their decomposition into the tensor product of an irreducibleU
q
(A
n
(1)
)-module and a bosonic Fock space were given. Here a general scheme for the wedge construction ofq-deformed Fock spaces using the theory of perfect crystals is presented.LetU
q
(g) be a quantum affine algebra. LetV be a finite-dimensionalU
q
(g)-module with a perfect crystal base of levell. LetV
aff V z,z
–1] be the affinization ofV, with crystal base (L
aff,B
aff). The wedge spaceV
aff V
aff is defined as the quotient ofV
aff V
aff by the subspace generated by the action ofU
q
(g) z
a
z
b
+z
b
z
a
]a,b onv v (v an extremal vector). The wedge space r
V
aff (r ) is defined similarly. Normally ordered wedges are defined by using the energy functionH :B
aff B
aff . Under certain assumptions, it is proved that normally ordered wedges form a base of r
V
aff.Aq-deformed Fock space is defined as the inductive limit of r
V
aff asr , taken 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 the Fock space has the structure of an integrableU
q
(g)-module. An action of the bosons, which commute with theU
q
(g)-action, is given on the Fock space. It induces the decomposition of theq-deformed Fock space into the tensor product of an irreducibleU
q
(g)-module and a bosonic Fock space.As examples, Fock spaces for typesA
2n
(2)
,B
n
(1)
,A
2n
–1/(2)
,D
n
(1)
andD
n
+1/(2)
at level 1 andA
1
(1)
at levelk are constructed. The commutation relations of the bosons in each of these cases are calculated, using two point functions of vertex operators. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|