We define the entropy of a graded algebra A=⊕nAn by limsupn→∞dimAn. This is related to the notion of entropy in symbolic dynamics, and could serve as a natural dimension concept for algebras with exponential growth. We study the entropy of quotients and subalgebras of free associative algebras and free Lie algebras. We also study the behavior of the entropy function under free products, and obtain several characterizations of free algebras in terms of entropy.
Bibliographical noteFunding Information:
During the preparation of this paper we had fruitful discussions with several people. We are grateful to B. Weiss, E. Lindenstrauss, and L. Small for their useful comments and suggestions. A. Shalev is grateful to The Australian National University for its kind hospitality while some of this work was carried out. A. Shalev also acknowledges the support of the BSF Grant No. 96-00101.
- Free Lie algebras
- Free associative algebras
- Graded Lie algebras
- Graded associative algebras
- Hilbert series
- Symbolic dynamics