Some of A.I. Shirshov’s works

  • Vladislav K. Kharchenko
Part of the Contemporary Mathematicians book series (CM)


In his first published paper “Subalgebras of free Lie algebras” A.I. Shirshov proved for Lie algebras an analog of the famous Nielsen-Schreier theorem: every subalgebra of a free Lie algebra is free. Three years later this theorem was independently proved and extended to restricted Lie algebras by E. Witt [38]. Much later this result was generalized to Lie superalgebras (A.S. Shtern [29]), and to colored Lie superalgebras (A.A. Mikhalev [20, 21, 22]). These results went through further development in the field of quantum algebra as follows. The Shirshov-Witt theorem for Lie algebras over fields of characteristic zero admits an equivalent formulation in terms of a free associative algebra: Every Hopf subalgebra of a free algebra k<y i > with the coproduct defined by Δ (y i )=y i ⊗1+1⊗y i is free. If we consider the free algebra as a braided Hopf algebra with a very special braiding (τ(y i y j )=p ij y j y i P ij P ji =1), then we get a reformulation of the Mikhalev-Shtern generalization as well. We may consider the free associative algebra k<V> as a braided Hopf algebra provided that V is a braided space with arbitrary braiding (not necessary invertible). In this setting the braided version of the Shirshov-Witt theorem takes the following form [12]: If a subalgebra \( U \subseteq k\left\langle V \right\rangle \) is a right categorical right coideal, that is \( \Delta U \subseteq U\underline \otimes k\left\langle V \right\rangle \), \( \tau \left( {k\left\langle V \right\rangle \otimes U} \right) \subseteq U \otimes k\left\langle V \right\rangle \), then U is a free subalgebra.


Hopf Algebra Commutative Ring Free Algebra Associative Ring Free Associative Algebra 
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© Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 2009

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  • Vladislav K. Kharchenko

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