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On the generalized Lie structure of associative algebras

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Abstract

We study the structure of Lie algebras in the categoryH MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA inH MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain.

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Moscow State University, 119899, Moscow, Russian Federation

    Y. Bahturin

  2. Department of Mathematics, California State University, 92407, San Bernardino, CA, USA

    D. Fischman

  3. Department of Mathematics, University of Southern California, 90089-1113, Los Angeles, CA, USA

    S. Montgomery

Authors
  1. Y. Bahturin
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  2. D. Fischman
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  3. S. Montgomery
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Corresponding author

Correspondence to Y. Bahturin.

Additional information

Dedicated to the memory of S. A. Amitsur

Supported by a Fulbright grant.

Supported by NSF grant DMS-9203375.

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Bahturin, Y., Fischman, D. & Montgomery, S. On the generalized Lie structure of associative algebras. Israel J. Math. 96, 27–48 (1996). https://doi.org/10.1007/BF02785532

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  • DOI: https://doi.org/10.1007/BF02785532

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