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Group identities on the units of algebraic algebras with applications to restricted enveloping algebras. (English) Zbl 1151.16034

An algebra \(A\) is called a GI-algebra if its group of units \(A^\times\) satisfies a group identity. The authors provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically generated GI-algebra locally finite?
For an algebra \(A\), we denote by \(B(A)\) its prime radical, by \(J(A)\) its Jacobson radical, and by \(Z(A)\) its center. Also, the set of all nilpotent elements of \(A\) will be denoted by \(N(A)\).
The following proposition is interesting. Proposition 1.1. Let \(A\) be an algebraic algebra over an infinite field \(F\) of characteristic \(p\geq 0\). Then the following conditions are equivalent: 1. The algebra \(A\) is a GI-algebra. 2. The group of units \(A^\times\) is solvable, in the case when \(p=0\), while \(A^\times\) satisfies a group identity of the form \((x,y)^{p^t}=1\) for some natural number \(t\), in the case when \(p>0\). 3. The algebra \(A\) satisfies a non-matrix polynomial identity. 4. The algebra \(A\) is Lie solvable, in the case when \(p=0\), while \(A\) satisfies a polynomial identity of the form \(([x,y]z)^{p^t}=0\) for some natural number \(t\), in the case when \(p>0\). – Furthermore, in this case, \(N(A)=B(A)\) is a locally nilpotent ideal of \(A\) and \(A/B(A)\) is both commutative and reduced.
In Section 2 essential results are the following.
Theorem 2.1. Let \(A\) be an algebraic algebra over a field \(F\) with \(|F|\geq 4\). Then \(A^\times\) is solvable if and only if all of the following conditions hold: \(A\) is Lie solvable, \(A/J(A)\) is commutative, and there exists a chain \(0=J_0\subseteq J_1\subseteq\cdots\subseteq J_m=J(A)\) of ideals of \(J(A)\) such that every factor \(J_i/J_{i-1}\) is the sum of commutative ideals of \(J_m/J_{i-1}\).
Theorem 2.2. Let \(A\) be an algebraic algebra over a perfect field \(F\neq F_2\). 1. The group of units \(A^\times\) is bounded Engel if and only if \(A\) is bounded Engel. In this case, \(N(A)\) is a locally nilpotent ideal such that \(A=Z(A)+N(A)\). 2. The group of umits \(A^\times\) is nilpotent if and only if \(A\) is Lie nilpotent. Furthermore, the corresponding nilpotency classes coincide.
Theorem 2.4. Let \(A\) be an algebra over a (finite) field of characteristic \(p>0\). Suppose \(L\subseteq A\) is a Lie subalgebra consisting of nilpotent elements. If either \(A^\times\) is solvable or bounded Engel then the associative subalgebra \(S\) generated by \(L\) is locally nilpotent.
The main result of the paper is the following theorem.
Theorem 3.1. Let \(L\) be a restricted Lie algebra over an infinite perfect field of characteristic \(p>0\). If \(L\) is algebraically generated and \(u(L)\) is a GI-algebra then \(u(L)\) is locally finite.
For its proof several lemmas are used. By using this theorem some interesting corollaries are obtained. – Finally the authors mention that the Kurosh problem for restricted enveloping algebras remains open: Problem 3.11. Is every algebraic restricted Lie algebra \(L\) locally finite?

MSC:

16U60 Units, groups of units (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
17B35 Universal enveloping (super)algebras
16S30 Universal enveloping algebras of Lie algebras
16P10 Finite rings and finite-dimensional associative algebras
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References:

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