which submodule of FG-module of a lie algebra $L$ will be determined I want to check that how we can find out a classical lie algebra like $D_4$ and $E_6$ are irreducible?
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4$\begingroup$ Please give us more information, for instance, how did this question arise, what do you already know. Otherwise this seems rather unmotivated past "I need to answer this question", which unfortunately makes it seem like homework. See also mathoverflow.net/help/how-to-ask $\endgroup$– David Roberts ♦Commented Nov 6, 2013 at 22:57
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1$\begingroup$ As David's comment suggests, your question is not yet well formulated. For instance, "irreducibility" of a Lie algebra here is not clearly defined. And of course there are problems about restricting from a large field to a very small one. $\endgroup$– Jim HumphreysCommented Nov 6, 2013 at 23:31
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$\begingroup$ i will improve my question as soon as possible. $\endgroup$– user118746Commented Nov 6, 2013 at 23:38
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$\begingroup$ This is not a clear question. Please clarify what you mean exactly. $\endgroup$– user42090Commented Nov 7, 2013 at 2:00
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To clarify what is going on in such special cases, it will help to have two specific references:
G.M.D. Hogeweij, Almost-classical Lie algebras. I, II. Nederl. Akad. Wetensch. = Indag. Math. 44 (1982), no. 4, 441–452, 453–460.
Gerhard Hiss, Die adjungierten Darstellungen der Chevalley-Gruppen. [The adjoint representations of the Chevalley groups] Arch. Math. (Basel) 42 (1984), no. 5, 408–416.
The first is based on Hogeweij's 1978 Utrecht thesis (and is his only publication), while the second was developed independently by Hiss early in his career. Hogeweij describes the ideal structure over an arbitrary field of the Lie algebra of a simple algebraic group of each possible type. Here there are typically a number of non-isomorphic groups ranging from universal to adjoint type, with possibly different Lie algebras over fields of prime characteristic. (For type $A_\ell$ this was worked out in my thesis.)
On the other hand, Hiss focuses on "Chevalley groups" in the original sense of Chevalley's 1955 construction, so the ambient algebraic groups are always of universal type. Then the "Chevalley" Lie algebra of each type can be described over an arbitrary field as a module for the adjoint action of the group. While Hiss developed his results independently, he became aware of the work of Hogeweij at some point.
Limiting discussion to the Lie algebras of the adjoint groups of type $D_4$ and $E_6$ over $\mathbb{F}_2$, it's made clear in the papers cited that the Lie algebra has 0 center for $E_6$ but has center of dimension 2 for $D_4$. The Lie algebra of type $E_6$ is in fact simple, and the adjoint representation is irreducible over $\mathbb{F}_2$. On the other hand, the Lie algebra of type $D_4$ is simple modulo its center, while the quotient affords an irreducible representation of the adjoint group over $\mathbb{F}_2$.
Hogeweij's full printed thesis has a convenient fold-out at the end displaying concisely all of the ideal structure for the Lie algebras attached to various groups of each Lie type, but I'm not sure how the results are presented in the published two-part paper.
The moral is that one has to work rather carefully with individual types, including isogeny types besides the adjoint groups, in order to get a full picture for each prime. Even so, the Chevalley construction, which starts with an integral form in a complex simple Lie algebra, yields nice results over any field including finite fields. Steinberg's more general construction is done using faithful representations, not just the adjoint representation, along with suitable $\mathbb{Z}$-forms.
answered Nov 7, 2013 at 1:05