The formal shift operator on the Yangian double

DOI10.1093/IMRN/RNAB026zbMATH Open1515.16030arXiv2008.10590MaRDI QIDQ6347650

Publication date: 24 August 2020

Abstract: Let

m a t h f r a k g

be a symmetrizable Kac-Moody algebra with associated Yangian

Y_{h} b a r m a t h f r a k g

and Yangian double

m a t h r m D Y_{h} b a r m a t h f r a k g

. An elementary result of fundamental importance to the theory of Yangians is that, for each

c i n m a t h b b C

, there is an automorphism

a u_{c}

of

Y_{h} b a r m a t h f r a k g

corresponding to the translation

t m a p s t o t + c

of the complex plane. Replacing

c

by a formal parameter

z

yields the so-called formal shift homomorphism

a u_{z}

from

Y_{h} b a r m a t h f r a k g

to the polynomial algebra

Y_{h} b a r m a t h f r a k g [z]

. We prove that

a u_{z}

uniquely extends to an algebra homomorphism

P h i_{z}

from the Yangian double

m a t h r m D Y_{h} b a r m a t h f r a k g

into the

h b a r

-adic closure of the algebra of Laurent series in

z^{- 1}

with coefficients in the Yangian

Y_{h} b a r m a t h f r a k g

. This induces, via evaluation at any point

c i n m a t h b b C^{i} m e s

, a homomorphism from

m a t h r m D Y_{h} b a r m a t h f r a k g

into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of

m a t h r m D Y_{h} b a r m a t h f r a k g

and

Y_{h} b a r m a t h f r a k g

and, as a corollary, we find that the Yangian

Y_{h} b a r m a t h f r a k g

can be realized as a degeneration of the Yangian double

m a t h r m D Y_{h} b a r m a t h f r a k g

. Using these results, we obtain a Poincar'{e}-Birkhoff-Witt theorem for

m a t h r m D Y_{h} b a r m a t h f r a k g

applicable when

m a t h f r a k g

is of finite type or of simply-laced affine type.

Mathematics Subject Classification ID

Deformations of associative rings (16S80) Graded rings and modules (associative rings and algebras) (16W50) Universal enveloping algebras of Lie algebras (16S30) Ring-theoretic aspects of quantum groups (16T20)

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