Independent systems of derivations and Lie algebra representations

Skryabin, Serge; Arslanov, M. M.; Parshin, A. N.; Shafarevich, Igor R.

doi:10.1515/9783110889550-012

Cited by 5 publications

(8 citation statements)

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“…Lemma 3.2 now implies that those ψ α s in (3.6) and (3.7) are nilpotent. We may use the same inductive arguments found in the proof of [18,Lemma 4.5] The proof is now complete.…”

Section: P (I)mentioning

confidence: 87%

“…As we showed in the proof of Proposition 3.3, (ii) is a direct consequence of (i). By Proposition 3.3 we only need to prove that M is a R-submodule of M. We will make use of the strategy that Skryabin proposed for W(m; n) in [18].…”

Section: P (I)mentioning

confidence: 99%

“…3.1. The (R, L)-mod and the category C. In [18] Skryabin introduced the category C for the study of representations of the generalized Jacobson-Witt algebra. In this section we shall extend this category to the Hamiltonian algebra H(2r; n).…”

Section: Preliminariesmentioning

confidence: 99%

“…In a generalization of Shen's work, Skryabin studied representations of L more conceptually in [18]. Shen's mixed product combining two modules of R and L is extended to be a so-called (R, L)-module structure in the more general setting of commutative algebras and their differential systems.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is a continuation of our previous work (see [17,24]). Recall that Skryabin first introduced the category C for the generalized Jacobson-Witt algebra W(m; n) in [18]. Recall that W(m; n) 0 consists of 'differential operators' of degree equal to or greater than zero, that is, of the form m i=1 f i D i with f i having no constants for i = 1, .…”

Section: Introductionmentioning

confidence: 99%

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IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

Yao¹,

Shu²

2011

J. Aust. Math. Soc.

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Let L = H(2r; n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristic p > 2. In the generalized restricted Lie algebra setup, any irreducible representation of L corresponds uniquely to a (generalized) p-character χ. When the height of χ is no more than min{p ni − p ni−1 | i = 1, 2, . . . , 2r} − 2, the corresponding irreducible representations are proved to be induced from irreducible representations of the distinguished maximal subalgebra L 0 with the aid of an analogy of Skryabin's category C for the generalized Jacobson-Witt algebras and modulo finitely many exceptional cases. Since the exceptional simple modules have been classified, we can then give a full description of the irreducible representations with p-characters of height below this number.2010 Mathematics subject classification: primary 17B10; secondary 17B50, 17B70.

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“…Lemma 3.2 now implies that those ψ α s in (3.6) and (3.7) are nilpotent. We may use the same inductive arguments found in the proof of [18,Lemma 4.5] The proof is now complete.…”

Section: P (I)mentioning

confidence: 87%

Section: P (I)mentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

Yao¹,

Shu²

2011

J. Aust. Math. Soc.

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Actions of Monoidal Categories and Representations of Cartan Type Lie Algebras

Pei

Sheng²,

Tang³

et al. 2022

J. Inst. Math. Jussieu

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Using crossed homomorphisms, we show that the category of weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs). This generalises and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras and to obtain new representations for generalised Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

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