Gradings on Finite-Dimensional Simple Lie Algebras
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Gradings on Finite-Dimensional Simple Lie Algebras Mikhail Kochetov
Received: 19 November 2008 / Accepted: 19 November 2008 / Published online: 17 December 2008 © Springer Science+Business Media B.V. 2008
Abstract In this survey paper we present recent classification results for gradings by arbitrary groups on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from 2. We also describe the main tools that were used to obtain these results (in particular, the classification of group gradings on matrix algebras). Keywords Graded algebra · Simple Lie algebra · Grading · Involution Mathematics Subject Classification (2000) 17B70
1 Introduction Let A be an algebra (not necessarily associative) over a field F and let G be a group. We will usually use multiplicative notation for G, but for abelian groups we will sometimes switch to additive notation. Definition 1.1 A G-grading on A is a vector space decomposition A= Ag g∈G
such that Ag Ah ⊂ Agh
for all g, h ∈ G.
This paper is an extended and updated version of the talk given by the author at the workshop “Groups, Rings, Lie and Hopf Algebras, II” held at Bonne Bay Marine Station, Memorial University of Newfoundland, on August 13–17, 2007. The author acknowledges support by NSERC Discovery Grant # 341792-07. M. Kochetov () Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C5S7, Canada e-mail: [email protected]
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M. Kochetov
Ag is called the homogeneous component of degree g. The support of the G-grading is the set S = {g ∈ G | Ag = 0}. Example 1.2 The matrix algebra R = Mn (F) has a Z2 -grading associated to each block A B decomposition C D , with A ∈ Ml (F), D ∈ Mn−l (F): R0 =
A 0 0D
R1 =
and
0B C 0
.
More generally, if (g1 , . . . , gn ) is an n-tuple of elements in G, then we obtain a G-grading on R = Mn (F) by setting Rg = Span{Eij | gi−1 gj = g},
(1)
where Eij are the matrix units. Example 1.3 There is a Z2 × Z2 -grading on R = M2 (C) associated to the Pauli matrices σ3 =
1
0
0
−1
σ1 =
,
0
1
1
0
σ2 =
,
0
i
−i
0
.
Namely, we set R(0,0) =
0
0
α
0
γ
γ
0
,
R(0,1) =
α
R(1,0) =
,
R(1,1) =
β
0
0
−β
0
δ
−δ
0
,
(2) .
If F contains a primitive n-th root of unity ε, then we can define the following n × n matrices that generalize −σ3 and σ1 : ⎡
ε n−1
⎢ ⎢ 0 ⎢ ⎢ Xa = ⎢ ⎢ ... ⎢ ⎢ 0 ⎣ 0
0
0
... 0
ε n−2
0
... 0
0
0
...
0
0
... 0
ε
0
⎡
⎤
⎥ 0⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0⎥ ⎦
and
0
1 0
⎢ ⎢ 0 0 1 ⎢ ⎢ Xb = ⎢ ⎢... ⎢ ⎢ 0 0 0 ⎣
1
1
0 0
... 0 0
⎤
⎥ ... 0 0⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ... 0 1⎥ ⎦ ... 0 0
(3)
Since Xa Xb = εXb Xa and Xan = Xbn = I , the following is a Zn × Zn -grading on R = Mn (F): R(k,l) = Span{Xak Xbl }.
(4)
It turns out that any grading on Mn (F) by an abelian group G can be obtained by combining gradings of the form (1) and (4)—see Sect. 5.
Gradings on Lie algebras
103
Example 1.4 Let g be a finite-dimensional semisimple Lie algebra over C. Let h be a Cartan subalgebra.
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