Towards a cladistics of double Yangians and elliptic algebras*

Abstract

A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangians structures of dynamical type are in particular defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists. MSC number: 81R50, 17B37LAPTH-738/99 PAR-LPTHE 99-23 DTP-99-45 math.QA/9906189June 19991 Cladistics : a system of biological taxonomy that defines taxa uniquely by shared characteristics not found in ancestral groups and uses inferred evolutionary relationships to arrange taxa in a branching hierarchy such that all members of a given taxon have the same ancestors.

Elliptic quantum group Uq,p(sl̂Konno1 2009Journal of Geometry and Physics243430View full textAdd to dashboardCite

“…Here U q,x ( sl 2 ) is the dynamical quantum affine algebra suggested in [48]. However neither its generators nor the L operators have yet been given explicitly.…”

Radar research at The Pennsylvania State University Radar and Communications Laboratory

“…Let us define the dynamical affine algebra based on the RLL relations associated to a dynamical affine R-matrix. The following R-matrix can be derived [1,12] using the representation theory of U q ( sl 2 ) and is the degeneration at p → 0 of the elliptic R-matrix presented in [15,21]:…”

“…By Theorem 3.11, P V (u) uniquely determines d, the set of eigenvalues of {ψ n , ϕ −n } n≥0 . For each a j , let Ṽj = V (1) (q 2α j ) denote the U (R)-module from Corollary 3.17, with PHWV ṽj := v (1) 0 ⊗ 1 and let W = ⊗ j Ṽj . Evidently, W is a DPHWM with PHWV Ω := ⊗ j ṽj , satsifying the condition that q h Ω = q r Ω.…”

“…(vi) U q,x = U q,x ( sl 2 ) is constructed in [23] and its representation theory is the topic of this treatise. A definition based on RLL-relations is suggested, but not expounded, in [1]. The non-elliptic, non-dynamical and non-affine degenerations can be considered on the level of R-matrices, algebras, modules and intertwiners (6j-symbols).…”

Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

“…From Theorem 4.8, P V (u) determines the set of eigenvalues d of ψ k and φ −k (k ∈ Z ≥0 ) uniquely. Consider the representation V = V (1)…”