Fine gradings on $\mathfrak{e}_6$

Abstract

Abstract. There are fourteen fine gradings on the exceptional Lie algebra e 6 over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that they are all.

Weyl groups of some fine gradings one6

“…In this section we describe the six gradings by infinite universal grading groups different from the Cartan grading. Our descriptions will not be, in most cases, the same as those in [8], to adapt them to our study of symmetries. This makes it necessary to recall some properties of the quasitori inducing them, and, particularly, of the automorphisms involved.…”

“…Moreover, these fine gradings immediately provide bases with interesting properties, as proved in [7,Proposition 10]. In a particular case, [8,Lemma 1] shows that every element in any basis of homogeneous elements of a fine grading by a finite group is semisimple.…”

“…The main theorem in [8] gives the classification up to equivalence of the fine gradings on e 6 . It encloses the following table, which provides, for each fine grading, its universal grading group and the type of the grading.…”

“…The knowledge of such gradings (extracted from [8]) make us think that a good Z-grading as a starting point would be the one related to the following marked Dynkin diagram:…”

“…This technique is applied in [16] for obtaining all the graded contractions of the Lie algebra sl 3 (C) arising from the Pauli grading. Afterwards, the same job has been taken up in [12] and [13] for a variety of algebras: matrix algebras, octonion algebras, Albert algebras, and Lie algebras of types A, B, C and D. As g 2 is the derivation algebra of the octonion algebra, and f 4 is the derivation algebra of the Albert algebra, our goal will be to determine these Weyl groups for e 6 , the following simple Lie algebra in size, and the only one of the series E for which the fine gradings are completely classified up to equivalence [8].…”