On the combinatorics of the universal enveloping algebra \(\widehat{U}_h({{\mathfrak {sl}}}_2)\)

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Abstract

Using combinatorial methods we study the structural coefficients of the formal homogeneous universal enveloping algebra \(\widehat{U}_h({\mathfrak {sl}}_2) \) of the special linear algebra \( {\mathfrak {sl}}_2\) over a characteristic zero field. We provide explicit formulae for the product of generic elements in \( \widehat{U}_h({\mathfrak {sl}}_2),\) and construct combinatorial objects giving flesh to these formulae; in particular, we provide explicit formulae and combinatorial interpretations for the structural coefficients of divided power Poincaré–Birkhoff–Witt basis.

Keywords

Universal enveloping algebras Poincaré–Birkoff–Witt Combinatorics 

Mathematics Subject Classification

17B45 16S30 05A19 

Notes

Acknowledgements

E. Salamanca was partially supported by a “Young Researcher” − COLCIENCIAS grant.

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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2018

Authors and Affiliations

  1. 1.Facultad de Ciencias, Escuela de MatemáticasUniversidad Nacional de Colombia - Sede MedellínMedellínColombia
  2. 2.FaMAF-CIEM (CONICET)Universidad Nacional de CórdobaCórdobaArgentina

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