Bokut, L.A., Latyshev, V., Shestakov, I., Zelmanov, E. (Eds.)
Translated by Kotchetov, M.V., Bremner, M.
2009, VII, 242 p.
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English translations of Shirshov’s works, which were published only in Russian
Due to Shirshov’s future leading and unique handling of algebraic problems, the results of his work are still of high interest for current research in algebra
This book presents translations of selected works of the famous Russian mathematician A.I. Shirshov (1921–1981). He was a pioneer in several directions of associative, Lie, Jordan, and alternative algebras, as well as groups and projective planes. His name is associated with notions and results on Gröbner-Shirshov bases, the Composition-Diamond Lemma, the Shirshov-Witt Theorem, the Lazard-Shirshov elimination process, Shirshov’s Height Theorem, Lyndon-Shirshov words, Hall-Shirshov bases, Shirshov’s theorem on the Kurosh problem for alternative and Jordan algebras, and Shirshov’s theorem on the speciality of Jordan algebras with two generators. Shirshov’s ideas were used by his student Efim Zelmanov for the solution of the Restricted Burnside Problem. Several famous algebraists provide in this book detailed comments on the impact of Shirshov’s work on current algebra.
Publications of A.I. Shirshov: Subalgebras of free Lie algebras.- On the representation of Lie rings in associative rings.- Subalgebras of free commutative and free anticommutative algebras.- On special J-rings.- Some theorems on embedding of rings.- On some nonassociative nil-rings and algebraic algebras.- On rings with identical relations.- On free Lie rings.- On a problem of Levitzki.- Some problems in the theory of rings that are nearly associative.- Some algorithmic problems for e-algebras.- Some algorithmic problems for Lie algebras.- On a hypothesis in the theory of Lie algebras.- On the bases of a free Lie algebra.- On some groups which are nearly Engel.- On some identical relations for algebras.- On some positively definable varieties of groups.- On the definition of the binary-Lie property.- On the theory of projective planes (with A.A. Nikitin). Commentaries: Perspectives on Shirshov’s Height Theorem.- On Shirshov’s Papers for Lie Algebra.- Some of A.I. Shirshov’s Works.- Comments on Shirshov’s Height Theorem.- Shirshov’s Works on Alternative and Jordan Algebras.- A Word about the Teacher