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On R-matrix quantization of formal loop groups. (English) Zbl 0674.17002
Group theoretical methods in physics, Proc. 3rd Semin., Yurmala/USSR 1985, Vol. 2, 161-180 (1986).
[For the entire collection see Zbl 0656.00014.]
Let \({\mathfrak g}={\mathfrak gl}_ N[t]\) be the Lie algebra of polynomial maps \({\mathbb{C}}\to {\mathfrak gl}_ N({\mathbb{C}})\) and consider the Lie algebra \({\mathfrak g}^{(k)}={\mathfrak g}/t^ k{\mathfrak g}\). The author constructs, by the R-matrix method, a deformation \(A^{(k)}\) of universal enveloping algebra of \({\mathfrak g}^{(k)}\). This is a finite-dimensional analogue of the Yangian of \({\mathfrak gl}_ N({\mathbb{C}})\). The centre of \(A^{(k)}\) is described in terms of Hecke algebras, and a class of irreducible finite- dimensional representations of \(A^{(k)}\) is constructed; this class is conjectured to be complete.
Reviewer: A.N.Pressley

MSC:
17B65 Infinite-dimensional Lie (super)algebras
58H15 Deformations of general structures on manifolds
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B35 Universal enveloping (super)algebras