Subalgebras and automorphisms of Lie algebras of cartan type

Rudakov, A N

doi:10.1007/bf01077325

Cited by 13 publications

(5 citation statements)

References 2 publications

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The natural filtrations of the infinite-dimensional modular odd hamiltonian superalgebras are proved to be invariant under automorphism. In nonmodular case, the natural filtrations of the infinite-dimensional Lie algebras X(m) and b X XðmÞ where X ¼ W; S; H or K were proved to be invariant in Rudakov (1986). We are thereby able to obtain an intrinsic characterization of odd hamiltonian superalgebras and establish a property of automorphisms of these Lie superalgebras.

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confidence: 90%

Infinite-Dimensional Modular Odd Hamiltonian Superalgebras

Liu

Zhang

2004

Communications in Algebra

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The natural filtrations of the infinite-dimensional modular odd hamiltonian superalgebras are proved to be invariant under automorphism. The proof involves the investigation of the ad-nilpotent elements of the even part, and the determination of the subalgebras generated by certain ad-nilpotent elements. We are thereby able to obtain an intrinsic characterization of odd hamiltonian superalgebras and establish a property of automorphisms of these Lie superalgebras. ), respectively. Cartan-type Lie algebras and Lie superalgebras possess natural filtration structures. In the modular Cartan-type Lie algebra case, Kostrikin and Shafarevic (1969) proved that the natural filtration of X(m : 1) is invariant under the automorphism group AutX(m : 1), where X ¼ W; S; H or K; later, Kac (1974) obtained the same conclusion for all other finite-dimensional Cartan-type Lie algebras and Jin (1992), using ad-nilpotent elements, obtained the same conclusion for infinite-dimensional Cartan-type Lie algebras. In nonmodular case, the natural filtrations of the infinite-dimensional Lie algebras X(m) and b X XðmÞ where X ¼ W; S; H or K were proved to be invariant in Rudakov (1986). In the case of finite-dimensional modular Lie superalgebras of Cartan type, the natural filtration of the hamiltonian superalgebra is proved to be invariant under automorphisms, by means of the minimal dimension of image spaces (see Zhang and Fu, 2002); similar results for the generalized Witt superalgebra and the special superalgebra are obtained in Zhang and Nan (1998) using ad-nilpotent elements. The invariance of natural filtrations of Cartan-type Lie algebras or Lie superalgebras provides a useful method of determining intrinsic properties and the automorphism groups (see ShenSec. 1 by recalling some definitions and collecting some useful facts about modular Lie superalgebras HO. In Sec. 2, we first establish some computational lemmas about p-adic numbers and p-adic matrices, which will be employed to prove that every element in HO 1 is ad-nilpotent. We next discuss ad-nilpotent elements of HO ½À1 [ HO ½0 and determine some subalgebras generated by ad-nilpotent elements. In Sec. 3, the results obtained in Sec. 2 will be employed to prove that L 0 is invariant under j defined above, and so is the natural filtration ðHO i Þ i!À1 . Applying the invariance of the natural filtration, we obtain an intrinsic characterization of HO and a property of Aut(HO). Infinite-Dimensional Modular Odd Hamiltonian Superalgebras

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confidence: 90%

Infinite-Dimensional Modular Odd Hamiltonian Superalgebras

Liu

Zhang

2004

Communications in Algebra

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“…Therefore, G c n = G n . Theorem 1.1 was announced in [8] where a sketch of the proof is given based on a study of certain Lie subalgebras of div 0 n of finite codimension. Our proof is based on completely different ideas.…”

Section: Introductionmentioning

confidence: 99%

“…The above theorem was announced in [8] where a sketch of a proof is given, it can also be deduced from [10]. A proof of the above theorem is given in [9] and in ( [7], Theorem 3.6), a different approach and a short proof is given in [2].…”

Section: Introductionmentioning

confidence: 99%

The groups of automorphisms of the Lie algebras of polynomial vector fields with zero or constant divergence

Bavula

2016

Communications in Algebra

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Let Pn = K[x1, . . . , xn] be a polynomial algebra over a field K of characteristic zero and div 0 n (respectively, div c n ) be the Lie algebra of derivations of Pn with zero (respectively, constant) divergence. We prove that AutLie(div 0 n ) ≃ Aut K−alg (Pn) (n ≥ 2) and AutLie(div c n ) ≃ Aut K−alg (Pn). The Lie algebra div c n is a maximal Lie subalgebra of DerK (Pn). Minimal finite sets of generators are found for the Lie algebras div 0 n and div c n .

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“…Theorem 1.1 was announced in [3] where a sketch of the proof is given based on a study of certain Lie subalgebras of div 0 n of finite codimension. The proof in [1] is based on completely different ideas.…”

Section: Introductionmentioning

confidence: 99%