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The meromorphic R-matrix of the Yangian | | Sachin Gautam
; Valerio Toledano-Laredo
; Curtis Wendlandt
; | | Date: |
8 Jul 2019 | | Abstract: | Let g be a complex semisimple Lie algebra and Yg the Yangian of g. The main
goal of this paper is to clarify the analytic nature of Drinfeld’s universal
R-matrix of Yg. It is known that the radius of convergence of R(s) on the
tensor product of two finite-dimensional representations U,V is generally zero.
If U,V are irreducible, Drinfeld proved that R_{U,V}(s) factors as
R^rat_{U,V}(s) rho_{U,V}(s), where rho_{U,V} is a generally divergent formal
power series in s^{-1}, and R^rat_{U,V} a rational function of s.
We prove that such a factorisation exists for any U,V, where rho_{U,V}(s) is
an endomorphism of U(x)V which intertwines the action of Yg. We prove, however,
that no such factorisation exists which is natural in U,V, and satisfies the
cabling relations. Equivalently, the universal R-matrix does not give rise to a
rational commutativity constraint on the category of finite-dimensional
representations of Yg.
We construct instead two meromorphic commutativity constraints R^pm_{U,V},
which are related by a unitarity condition. We show that each possesses an
asymptotic expansion as s tends to infinity, with pm Re(s)>0, which has the
same formal properties as Drinfeld’s R_{U,V}(s), and therefore coincides with
the latter by uniqueness. In particular, we give an alternative, constructive
proof of the existence of the universal R-matrix of Yg.
Our construction relies on the Gauss decomposition R^-(s)R^0(s)R^+(s) of
R(s). The divergent abelian term R^0 was resummed on finite-dimensional
representations by the first two authors in arXiv:1403.5251. The main
ingredient of the present paper is the explicit construction of R^+(s),R^-(s).
We prove that they are rational functions on finite-dimensional
representations, and that they intertwine the coproduct of Yg and the deformed
Drinfeld coproduct introduced in arXiv:1403.5251. | | Source: | arXiv, 1907.3525 | | Services: | Forum | Review | PDF | Favorites | |
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