The hopf algebra rep Uqglˆ∞

• Edward Frenkel ^[1] ; Evgeny Mukhin
  1. [1] Department of Mathematics, University of California, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 8, Nº. 4, 2002, págs. 537-635
• Idioma: inglés
• DOI: 10.1007/bf02637313
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• Resumen

  • We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of UqglˆN in the limitN→∞. The resulting Hopf algebra Rep Uqglˆ∞ is a tensor product of its Hopf subalgebras Repa Uqglˆ∞ ,a ∈ ℂ×/q2ℤ. Whenq is generic (resp.,q 2 is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep a Uqglˆ∞ and the algebra of regular functions on the prounipotent proalgebraic group SL˜∞¯¯¯¯ (resp., GL˜l¯ ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep a Uqglˆ∞ spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of GL˜l¯ considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep a Uqglˆ∞ and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.