Lie gradings on associative algebras.

*(English)*Zbl 1207.17037This work applies the methods of functional identities to the study of group gradings on simple Lie algebras. The classical results on Lie maps of algebraic structures related to associative algebras finds here a counterpart in which group gradings on Lie algebras coming from associative ones are related to associative group gradings. Though some classical results on Lie maps deal with simple or prime rings, some recent advances in the matter allow to include in the study much general classes of rings. The key point to relate group gradings with Lie maps is the fact that the existence of a group grading on a Lie \(F\)-algebra \(L\) (with grading group \(G\)) is equivalent to the existence of a homomorphism of Lie algebras \(\rho: L\to L\otimes H\) where \(H=FG\) is the group algebra, so that (1) \((\rho\otimes\text{id}_H)\rho=(\text{id}_H\otimes\Delta)\rho\) and (2) \((\text{id}_L\otimes\epsilon)\rho=\text{id}_L\), where \(\Delta: H\to H\times H\), is the co-multiplication of the Hopf algebra \(H\) and \(\epsilon: H\to F\) its co-unit (as it is well-known \(H\) is a Hopf algebra for \(\Delta(g):=g\otimes g\), \(\epsilon(g):=1\) and co-inverse \(s(g):=g^{-1}\) for any \(g\in G\)). For instance if \(L=\oplus_{g\in G}L_g\) then above map \(\rho\) is given by \(\rho(a_g)=a_g\otimes g\) for \(a_g\in L_g\). The map \(\rho\) is a comodule action or more precisely \(L\) is said to be a (right) \(H\)-comodule.

If \(L\) is a Lie subalgebra generating an associative algebra \(A\), then it may happen that \(\rho: L\to L\otimes H\) can be extended to a comodule action \(\rho: A\to A\otimes H\). Thus the corresponding grading of \(L\) could be extended to a group grading on \(A\). The work under review uses the so called functional identities techniques of M. Brešar, M. A. Chebotar and W. S. Martindale to deal with this extension problem.

The fundamental results in the paper, Theorems 6.8, 7.1 and 8.4 show that under certain conditions, a grading on a Lie algebra \(L\) is induced by an associative or an involution grading of an associative algebra \(A\) generated by \(L\).

If \(L\) is a Lie subalgebra generating an associative algebra \(A\), then it may happen that \(\rho: L\to L\otimes H\) can be extended to a comodule action \(\rho: A\to A\otimes H\). Thus the corresponding grading of \(L\) could be extended to a group grading on \(A\). The work under review uses the so called functional identities techniques of M. Brešar, M. A. Chebotar and W. S. Martindale to deal with this extension problem.

The fundamental results in the paper, Theorems 6.8, 7.1 and 8.4 show that under certain conditions, a grading on a Lie algebra \(L\) is induced by an associative or an involution grading of an associative algebra \(A\) generated by \(L\).

Reviewer: Candido Martín González (Málaga)

##### MSC:

17B70 | Graded Lie (super)algebras |

16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |

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\textit{Y. Bahturin} and \textit{M. Brešar}, J. Algebra 321, No. 1, 264--283 (2009; Zbl 1207.17037)

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##### References:

[1] | Bahturin, Yu., Basic structures of modern algebra, (1991), Kluwer AP |

[2] | Yu. Bahturin, M. Kochetov, Group gradings on simple Lie algebras of type A in positive characteristic, submitted for publication |

[3] | Yu. Bahturin, M. Kochetov, S. Montgomery, Gradings on simple Lie algebras of type A in positive characteristic, Proc. Amer. Math. Soc., in press · Zbl 1168.17016 |

[4] | Bahturin, Yu.; Shestakov, I.; Zaicev, M., Gradings on simple Jordan and Lie algebras, J. algebra, 283, 849-868, (2005) · Zbl 1066.17018 |

[5] | Bahturin, Yu.; Zaicev, M., Graded algebras and graded identities, (), 101-139 · Zbl 1053.16032 |

[6] | Bahturin, Yu.; Zaicev, M., Gradings on simple Lie algebras of type “A”, J. Lie theory, 16, 719-742, (2006) · Zbl 1141.17006 |

[7] | Bahturin, Yu.; Zaicev, M., Involutions on graded matrix algebras, J. algebra, 315, 527-540, (2007) · Zbl 1129.16033 |

[8] | Beidar, K.I.; Brešar, M.; Chebotar, M.A., Functional identities revised: the fractional and the strong degree, Comm. algebra, 30, 935-969, (2002) · Zbl 1015.16028 |

[9] | Beidar, K.I.; Brešar, M.; Chebotar, M.A.; Martindale, W.S., On Herstein’s Lie map conjectures, III, J. algebra, 249, 59-94, (2002) · Zbl 1019.16021 |

[10] | Beidar, K.I.; Martindale, W.S.; Mikhalev, A.V., Rings with generalized identities, (1996), Marcel Dekker, Inc. · Zbl 0847.16001 |

[11] | Bergen, J.; Herstein, I.N.; Kerr, J.W., Lie ideals and derivations of prime rings, J. algebra, 71, 259-267, (1981) · Zbl 0463.16023 |

[12] | Brešar, M.; Chebotar, M.A.; Martindale, W.S., Functional identities, (2007), Birkhäuser Verlag · Zbl 1132.16001 |

[13] | Brešar, M.; Kissin, E.; Shulman, V., Lie ideals: from pure algebra to \(\operatorname{C}^\ast\)-algebras, J. reine angew. math., 623, 73-121, (2008) · Zbl 1163.46033 |

[14] | Herstein, I.N., Topics in ring theory, (1969), University of Chicago Press · Zbl 0232.16001 |

[15] | Montgomery, S., Hopf algebras and their actions on rings, CBMS reg. conf. ser. math., vol. 82, (1993), American Mathematical Society Providence, RI · Zbl 0804.16041 |

[16] | Patera, J.; Zassenhaus, H., On Lie gradings. I, Linear algebra appl., 112, 87-159, (1989) · Zbl 0675.17001 |

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