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Macdonald Polynomials from Sklyanin Algebras: A Conceptual Basis for the $p$-Adics-Quantum Group Connection | | Peter G. O. Freund
; Anton V. Zabrodin
; | | Date: |
24 Oct 1991 | | Journal: | Commun.Math.Phys. 147 (1992) 277-294 | | Subject: | High Energy Physics - Theory; Quantum Algebra | hep-th math.QA | | Abstract: | We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin’s two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate’’ between the zonal spherical functions of related real and $p$--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$--Baxter models. We show that in the $n air infty$ limit, the Jost function for the scattering of {em first} level excitations in the $Z_n$--Baxter model coincides with the Harish--Chandra--like $c$--function constructed from the Macdonald polynomials associated to the root system $A_1$. The partition function of the $Z_2$--Baxter model itself is also expressed in terms of this Macdonald--Harish--Chandra $c$--function, albeit in a less simple way. We relate the two parameters $q$ and $t$ of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the $p$--adic ``regimes’’ in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of ``$q$--deforming’’ Euler products. | | Source: | arXiv, hep-th/9110066 | | Services: | Forum | Review | PDF | Favorites | |
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