Irreducible Representations of the Generalized Jacobson-Witt Algebras

Shu, Bin; Yao, Yu-Feng

doi:10.1142/s1005386712000041

Cited by 10 publications

(6 citation statements)

References 12 publications

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“…Skryabin's C-module category has been extended to the case of special Lie algebras of Cartan type by the authors (see [24]). 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].…”

Section: Introductionsupporting

confidence: 52%

“…. , m. In the generalized restricted Lie algebra setup, the 'modified' induced modules for W(m; n) (induced from 'twist' modules of the distinguished maximal subalgebra W(m; n) 0 ) turn out to be objects of the category C (see [17]). The category C is described based on the understanding that Cartan type Lie algebras are Lie algebras of differential operators on the divided power algebras A(m; n).…”

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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Section: Introductionsupporting

confidence: 52%

Section: Introductionmentioning

confidence: 99%

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

Yao¹,

Shu²

2011

J. Aust. Math. Soc.

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“…Then we have the following direct corollary. Proof When the character χ satisfies the condition stated in the corollary, all irreducible non-exceptional U χ (L)-modules are baby Kac modules by [5][6][7]13,20] (see also [15,18,19]). Therefore, the statement is a direct consequence of Corollary 1.…”

Section: More General Resultsmentioning

confidence: 99%

Support varieties of semisimple-character representations for Cartan type Lie algebras

Yao

Shu

2011

Front. Math. China

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“…Roughly speaking, Lie(G) is actually the maximal distinguished filtered subalgebra L 0 . To a large extent, the study of irreducible representations of L can be reduced to those of L 0 (see [28], [31], [32], etc.). Such a G of the form G ⋉U is called semi-reductive in [25].…”

Section: Introductionmentioning

confidence: 99%

On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

Shu¹,

Xue²,

Yao³

2021

Preprint

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This is a sequel to [25] and [30]. Associated with G := GL n and its rational representation (ρ, M ) over an algebraically closed filed k, we define an enhanced algebraic group G := G ⋉ ρ M which is a product variety GL n × M , endowed with an enhanced cross product. In this paper, we first show that the nilpotent cone N := N(g) of the enhanced Lie algebra g := Lie(G) has finite nilpotent orbits under adjoint G-action if and only if up to tensors with one-dimensional modules, M is isomorphic to one of the three kinds of modules: (i) a one-dimensional module, (ii) the natural module k n , (iii) the linear dual of k n when n > 2; and M is an irreducible module of dimension not bigger than 3 when n = 2. We then investigate the geometry of enhanced nilpotent orbits when the finiteness occurs. Our focus is on the enhanced group G = GL(V ) ⋉ η V with the natural representation (η, V ) of GL(V ), for which we give a precise classification of finite nilpotent orbits via a finite set e P n of so-called enhanced partitions of n = dim V , then give a precise description of the closures of enhanced nilpotent orbits via constructing so-called enhanced flag varieties. Finally, the G-equivariant intersection cohomology decomposition on the nilpotent cone of g along the closures of nilpotent orbits is established.

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Irreducible Representations of the Generalized Jacobson-Witt Algebras

Cited by 10 publications

References 12 publications

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

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

Support varieties of semisimple-character representations for Cartan type Lie algebras

On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

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