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.
Author(s): Leonid A. Bokut, Victor Latyshev, Ivan Shestakov, Efim Zelmanov, Murray Bremner, Mikhail V. Kotchetov
Edition: 1
Year: 2009
Language: English
Pages: 235
Cover......Page 1
Selected Works
of
A.I. Shirshov......Page 5
Contemporary Mathematicians......Page 4
ISBN 3764388579......Page 6
Contents......Page 7
Preface......Page 9
Publications of
A.I. Shirshov......Page 10
Subalgebras of Free Lie Algebras......Page 11
On the Representation of Lie Rings
in Associative Rings......Page 22
Subalgebras of Free Commutative and
Free Anticommutative Algebras......Page 25
On Special J-rings......Page 32
Some Theorems on Embedding of Rings......Page 52
On some Nonassociative Nil-rings
and Algebraic Algebras......Page 60
On Rings with Identical Relations......Page 74
On Free Lie Rings......Page 81
On a Problem of Levitzki......Page 92
Some Problems in the Theory of Rings
that are Nearly Associative......Page 95
On the Bases of a Free Lie Algebra......Page 114
Some Algorithmic Problems for ε-algebras......Page 120
Some Algorithmic Problems for Lie Algebras......Page 126
On a Hypothesis in the Theory of Lie Alge......Page 132
On Some Groups which are Nearly Engel......Page 137
On Some Identical Relations for Algebras......Page 148
On Some Positively Definable Varieties
of Groups......Page 152
On the Definition of the Binary-Lie Property......Page 155
On the Theory of Projective Planes......Page 158
Commentaries......Page 180
Perspectives on Shirshov’s Height Theorem......Page 181
On Shirshov’s Papers for Lie Algebras......Page 199
Some of A.I. Shirshov’s Works......Page 212
Comments on Shirshov’s Height Theorem......Page 217
A.I. Shirshov’s Works
on Alternative and Jordan Algebras......Page 224
A Word about the Teacher......Page 232
Indication of Sources......Page 234
