Weyl groups of some fine gradings on \(\mathfrak{e}_6\)
exceptional Lie algebra \(\mathfrak{e}_6\)
The Weyl group of fine gradings with infinite universal grading group, on the exceptional simple Lie algebra \(E_6\) are computed. The ground field is supposed to be algebraically closed and of characteristic \(0\). The \(E_6\) Lie algebra has (up to equivalence) \(11\) fine gradings. There are \(7\) of these gradings whose universal grading group is finite and their study, by J. Yu, is almost complete. Thus the authors work in this paper with the complementary set of equivalence classes of fine gradings (ruling out the Cartan grading). The authors use a list of models suitable for their purposes. The list begins with the unified Tit's construction but includes some other models which, as far as this reviewer knows, are new. The constructions involve composition algebras, symmetric composition algebras and exceptional Jordan algebras. The six gradings under study are described in section 4. Roughly speaking they are: an inner \(\mathbb Z_2^3\times\mathbb Z^2\) grading denoted \(\Gamma_1\) and an outer \(\mathbb Z_2^3\times\mathbb Z^2\) grading denoted \(\Gamma_6\); a \(\mathbb Z_2^4\times\mathbb Z\) grading denoted \(\Gamma_2\); a \(\mathbb Z_2\times\mathbb Z^4\) grading denoted \(\Gamma_3\); a \(\mathbb Z_3^2\times\mathbb Z^2\) grading \(\Gamma_4\); and a \(\mathbb Z_2^5\times\mathbb Z\) grading \(\Gamma_5\). The Weyl groups are NEWLINE\[NEWLINEW(\Gamma_1)=\text{Mat}_{3\times 2}(\mathbb Z_2)\rtimes(GL_3(\mathbb Z_2)\times D_3)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_2)=\mathbb Z_2^4\rtimes(GL_3(\mathbb Z_2)\times \mathbb Z_2)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_3)=\mathbb Z_2^2\rtimes((\mathbb Z_2^3\rtimes S_4)\rtimes S_3)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_4)=\mathbb Z_3^2\rtimes(GL_2(\mathbb Z_3)\times D_6)NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_5)=(SP_4(\mathbb Z_2)\times\mathbb Z_2)\ltimes \mathbb Z_2^5NEWLINE\]NEWLINE NEWLINE\[NEWLINEW(\Gamma_6)=\mathbb Z_2^4\rtimes((\mathbb Z_2^2\rtimes S_3)\times (\mathbb Z_2^2\rtimes\mathbb Z_2)).NEWLINE\]
scientific article; zbMATH DE number 6339761
scientific article