Group Gradings on Finitary Simple Lie Algebras

Abstract

<jats:p>We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.</jats:p>

Fine gradings on $\mathfrak{e}_6$

“…There are four fine gradings on the algebra sl(3), with grading groups Z 2 , Z × Z 2 , Z 3 2 , Z 2 3 ; that is, there are four MAD-groups of Aut(sl(3)) ∼ = PSL(3) ⋊ Z 2 , but only two of them are inner, produced by quasitori of PSL(3), namely, Z 2 3 and a two-dimensional torus. (This result can be concluded from [9], but the gradings are explicitly computed in [25].) Hence the only possibilities for A are Q 3 = P 3 × P 4 and Q 4 = P 3 × S α,β .…”

“…On the other hand, F 3 , F 4 breaks each copy sl(V i ) of L0 in 8 pieces of dimension one (the non-toral Z 2 3 -grading on a 2 usually called Pauli grading, see for instance [9]), so, j L (0,j,k,l) has dimension three if (k,l) = (0,0) and breaks into three onedimensional pieces when F 2 is applied. Therefore all the homogeneous components have dimension one, except the following cases:…”

“…For algebraically closed fields of characteristic zero, the gradings on the classical Lie algebras have been studied in [6,9,16,18] and the gradings in some exceptional ones, namely, f 4 and g 2 , in [14,15,13,7]. Lately, some authors have already studied the case of prime characteristic, [5] in the classical case (with the exception of d 4 ) and [20] in f 4 and g 2 .…”

Classification of involutions on graded-division simple real algebras

“…Involutions on graded-division finite-dimensional simple complex algebras are classified in [9, Propositions 2.51 and 2.53] (see also [4]). In this paper we solve the real case.…”

On the Structure of Graded Lie Triple Systems

“…Since then, many papers describing different physic models by means of graded Lie type structures have appeared, being remarkable the interest on these objects in the last years. For instance, in the case of Lie algebras, we can cite many works related to theory of strings, to color supergravity, to Walsh functions, to electroweak interactions or to gauge models [1,4,9,10,17,18,20,22,25,29,34]. In the case of Lie superalgebras, we can also cite several works modelling continuous suppersymmetry transformations between bosons and fermions or conformal field theory [3,5,19,26,31].…”