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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