The entropy of graded algebras

M. F. Newman; Csaba Schneider; Aner Shalev

doi:10.1006/jabr.1999.8053

The entropy of graded algebras

M. F. Newman^*, Csaba Schneider, Aner Shalev

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

18 Citations (Scopus)

Abstract

We define the entropy of a graded algebra A=⊕_nA_n by limsup_n→∞dimA_n. 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.

Original language	English
Pages (from-to)	85-100
Number of pages	16
Journal	Journal of Algebra
Volume	223
Issue number	1
DOIs	https://doi.org/10.1006/jabr.1999.8053
Publication status	Published - 1 Jan 2000

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keywords = "Entropy, Free Lie algebras, Free associative algebras, Graded Lie algebras, Graded associative algebras, Growth, Hilbert series, Symbolic dynamics",

author = "Newman, {M. F.} and Csaba Schneider and Aner Shalev",

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AU - Schneider, Csaba

AU - Shalev, Aner

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N2 - 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.

AB - 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.

KW - Entropy

KW - Free Lie algebras

KW - Free associative algebras

KW - Graded Lie algebras

KW - Graded associative algebras

KW - Growth

KW - Hilbert series

KW - Symbolic dynamics

UR - http://www.scopus.com/inward/record.url?scp=0033633207&partnerID=8YFLogxK

U2 - 10.1006/jabr.1999.8053

DO - 10.1006/jabr.1999.8053

M3 - Article

SN - 0021-8693

VL - 223

SP - 85

EP - 100

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -

The entropy of graded algebras

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