Selected Works of A.I. Shirshov

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.

Résumé
Anatolii Illarionovich Shirshov (19211981) was an outstanding Russian mat- maticianwhoseworksessentiallyin?uenced thetheoriesofassociative,Lie,Jordan and alternative rings. Many Shirshov's students and students of his students had a successful research career in mathematics. AnatoliiShirshovwasbornonthe8thofAugustof1921inthevillageKolyvan near Novosibirsk. Before the II World War he started to study mathematics at Tomsk university but then went to the front to ?ght as a volunteer. In 1946 he continued his study at Voroshilovgrad (now Lugansk) Pedagogical Institute and at the same time taught mathematics at a secondary school. In 1950 Shirshov was accepted as a graduate student at the Moscow State University under the supervision of A. G. Kurosh. In 1953 he has successfully defended his Candidate of Science thesis (analog of a Ph. D. ) Some problems in the theory of nonassociative rings and algebras and joined the Department of Higher Algebra at the Moscow State University. In 1958 Shirshov was awarded the Doctor of Science degree for the thesis On some classes of rings that are nearly associative. In 1960 Shirshov moved to Novosibirsk (at the invitations of S. L. Sobolev and A. I. Malcev) to become one of the founders of the new mathematical institute of the Academy of Sciences (now Sobolev Institute of Mathematics) and to help the formation of the new Novosibirsk State University. From 1960 to 1973 he was a deputy director of the Institute and till his last days he led the research in the theory of algebras at the Institute.

Contenu
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.- On the Bases of a Free Lie Algebra.- Some Algorithmic Problems for ?-algebras.- Some Algorithmic Problems for Lie Algebras.- On a Hypothesis in the Theory of Lie Algebras.- 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.- Commentaries.- Perspectives on Shirshov's Height Theorem.- On Shirshov's Papers for Lie Algebras.- Some of A.I. Shirshov's works.- Comments on Shirshov's Height Theorem.- A.I. Shirshov's Works on Alternative and Jordan Algebras.- A Word about the Teacher.