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

São Paulo Journal of Mathematical Sciences volume 13, pages 342–369 (2019)Cite this article

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.

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Acknowledgements

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

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Authors and Affiliations

  1. Facultad de Ciencias, Escuela de Matemáticas, Universidad Nacional de Colombia - Sede Medellín, Medellín, Colombia

    Rafael Díaz

  2. FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Córdoba, Argentina

    Edward Salamanca

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  1. Rafael Díaz
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  2. Edward Salamanca
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Correspondence to Rafael Díaz.

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Díaz, R., Salamanca, E. On the combinatorics of the universal enveloping algebra \(\widehat{U}_h({{\mathfrak {sl}}}_2)\). São Paulo J. Math. Sci. 13, 342–369 (2019). https://doi.org/10.1007/s40863-018-0088-x

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  • DOI: https://doi.org/10.1007/s40863-018-0088-x

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