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Twisted vertex representations of quantum affine algebras. (English) Zbl 0692.17006

We construct vertex representations of quantum affine algebras twisted by an automorphism of the Dynkin diagram, i.e., the quantum affine algebras \({\mathcal U}_ q(A_ n^{(2)})\) (n\(\geq 2)\), \({\mathcal U}_ q(D_ n^{(2)})\) (n\(\geq 3)\), \({\mathcal U}_ q(E_ 6^{(2)})\), \({\mathcal U}_ q(D_ 4^{(3)})\) introduced in general by V. G. Drinfel’d and M. Jimbo. This generalizes the earlier joint work of I. B. Frenkel and the author [Proc. Natl. Acad. Sci. USA 85, 9373-9377 (1988; Zbl 0662.17006)] for the untwisted affine algebras. It also provides a q-deformation of some non-trivial cases of Frenkel and J. Lepowsky’s works on ordinary twisted vertex operators.
We define a twisted q-Fock space based on Drinfel’d’s realization of quantum loop algebras. Then the vertex operators for the twisted quantum affine algebras are defined via some familiar exponential operators with replacing the integers by the q-integers, and the periodic conditions are imposed thereafter. We prove that this gives the basic representations by checking the vertex operators satisfying the required relations. Our approach is a q-deformed vertex operator calculus, which offers some distinct features from the ordinary counterpart, for example, the Serre relations for the quantum affine groups are more subtle and provided by the method of formal power series. Also the correlation functions are expressed by the q-analog of the rational functions, which deforms the classical rational functions with multiorder poles into the ones with only simple poles.
The proof in the paper when specialized at the identity automorphism of the Dynkin diagram also provides a detailed one for the untwisted case (op. cit.).
Reviewer: N.H.Jing

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 0662.17006
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References:

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