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Díaz, Rafael; Salamanca, Edward

On the combinatorics of the universal enveloping algebra \(\widehat{U}_h(\mathfrak{sl}_2)\). (English) Zbl 1505.17010

São Paulo J. Math. Sci. 13, No. 1, 342-369 (2019).
Summary: 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.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S30 Universal enveloping algebras of Lie algebras
05A19 Combinatorial identities, bijective combinatorics

Keywords:

universal enveloping algebras; Poincaré-Birkoff-Witt basis; combinatorics
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References:

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