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Lie gradings on associative algebras. (English) Zbl 1207.17037
This 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 $$L$$.

MSC:
 17B70 Graded Lie (super)algebras 16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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References:
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