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The algebra of skew primitive elements. (English. Russian original) Zbl 0917.16017

Algebra Logika 37, No. 2, 181-223 (1998); translation in Algebra Logic 37, No. 2, 101-126 (1998).
The author studies the algebraic structure of the set of skew primitive elements of some Hopf algebras. The multiplicative structure, as compared with the structure of the set of primitive elements, suffers a sufficient distortion and turns into a partial operation of variable (quantum) arity; in other words, it splits into a set of closely related partial operations of different arities. These operations are the subject of the paper.
The author focuses his attention on the Hopf algebras generated by semi-invariants relative to an abelian group \(G\) of group-like elements acting by conjugation. The author calls such Hopf algebras “character algebras”. Among them, for instance, are quantum enveloping Drinfeld-Jimbo algebras; \(G\)-universal enveloping algebras of Lie color superalgebras; the quantum plane and any Hopf algebra generated by skew primitive elements, if it has exactly \(n\) group-like pairwise commuting elements, and the ground field contains a primitive \(n\)th root of unity.
The author describes all unary “quantum” operations and all binary operations which are linear in one of the variables, and presents a necessary and sufficient condition for a nonzero \(n\)-linear operation to exist. This criterion is used for a detailed study of trilinear and quadrilinear operations. In addition, the author introduces the notion of a partial principal operation of variable arity in terms of which all quantum operations of degree \(\leq 4\) are expressible in case the ground field is of characteristic \(0\).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
08A55 Partial algebras
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