Universal enveloping of (modified) λ-differential Lie algebras

ABSTRACT

We introduce a new operator – a modified λ-differential operator – from a usual differential operator. We construct the free (modified) λ-differential $(\mathbf{k},\mathrm{\partial })$-algebra on a (modified) differential $(\mathbf{k},\mathrm{\partial })$-module, and apply it to erect the universal enveloping (modified) differential algebra of a (modified) λ-differential Lie $(\mathbf{k},\mathrm{\partial })$-algebra. The corresponding Poincaré-Birchoff-Witt theorems are obtained. The Wronskian envelope of a modified λ-differential Lie $(\mathbf{k}\mathbf{,}\mathbf{-}\lambda {\mathrm{i}\mathrm{d}}_{\mathbf{k}})$-algebra is also constructed.

Keywords:

• Lie algebra
• differential algebra
• universal enveloping algebra
• PBW theorem

2010 Mathematics Subject Classifications:

• 16S30
• 16S10
• 17B35
• 17B60
• 12H05

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments which improved the paper. Yi Zhang thanks China Scholarship Council for supporting him to visit Department of Mathematics at University of Southern California.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11771191).