2017/09/30 by Curtis Wendlandt
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structure #Algebraic structures and combinatorial models #Graded Lie algebra #Hopf algebra #Lie algebra #Representation theory of Hopf algebras #Surjective function #Universal enveloping algebra #Yangian #math-ph #math.MP #math.QA #math.RT #msc:17B37 #msc:81R10
paper · pdf · doi:10.1007/s00220-018-3227-4
published as Commun. Math. Phys. (2018) 363: 289 · 33 pages
openalex created_date 2017/10/06 · openalex publication_date 2018/08/25 · arxiv created 2018/08/31 · arxiv updated 2018/10/09 · openalex updated_date 2026/08/05
Starting from a finite-dimensional representation of the Yangian Y(\mathfrakg) for a simple Lie algebra \mathfrakg in Drinfeld's original presentation, we construct a Hopf algebra XI(\mathfrakg), called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific R-matrix R(u). We prove that there is a surjective Hopf algebra morphism XI(\mathfrakg)\twoheadrightarrow Y(\mathfrakg) whose kernel is generated as an ideal by the coefficients of a central matrix Z(u). When the underlying representation is irreducible, we show that this matrix becomes a grouplike central series, thereby making available a proof of a well-known theorem stated by Drinfeld in the 1980's. We then study in detail the algebraic structure of the extended Yangian, and prove several generalizations of results which are known to hold for Yangians associated to classical Lie algebras in their R-matrix presentations.