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Universal enveloping of (modified) \(\lambda\)-differential Lie algebras. (English) Zbl 1496.16025

This paper deals with somewhat natural generalizations of differential (associative or Lie) algebras, namely \(\lambda\)-differential (associative or Lie) algebras and modified \(\lambda\)-differential (associative or Lie) algebras. A \(\lambda\)-differential (associative or Lie) algebra, for a constant \(\lambda\), is roughly an associative or Lie algebra \(A\) with a linear endomorphism \(A\xrightarrow{d}A\) which satisfies a relation similar to the Leibniz rule, namely \(d(xy)=xd(y)+d(x)y+\lambda d(x)d(y)\), \(x,y\in A\) (where, when \(A\) is a Lie algebra, a concatenation such as \(xy\) should be read as a Lie bracket \([x,y]\)). In the “modified” version the term \(\lambda d(x)d(y)\) is replaced by the term \(\lambda xy\).
The existence and uniqueness (up to a unique isomorphism) results about free objects are obtained easily by the authors because \(\lambda\)-differential \((\Bbbk,\partial)\)-modules and algebras, and their “modified” versions are categories concretely equivalent to varieties of algebras in the sense of universal algebra so that each algebraic functor, that is, a functor which preserves the underlying sets, has a left adjoint. Therefore the authors focus on explicit constructions for these objects, which is far more interesting and more tricky (see, e.g., Theorem 3.5, p. 1114, Theorem 3.8, p. 1116).

MSC:

16S30 Universal enveloping algebras of Lie algebras
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
17B35 Universal enveloping (super)algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
12H05 Differential algebra
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