Takiff Algebras With Polynomial Rings of Symmetric Invariants

Panyushev, Dmitri I.; Yakimova, Oksana

doi:10.1007/s00031-019-09532-9

Cited by 8 publications

(5 citation statements)

References 27 publications

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“…This result was extended by Raïs and Tauvel [18] to all values of ℓ. More recently, Macedo and Savage [12] proved its multi-parameter generalization, while Panyushev and Yakimova [17] showed that this generalization remains valid for a wide class of Lie algebras g beyond simple Lie algebras.…”

Section: Introductionmentioning

confidence: 81%

Casimir elements and Sugawara operators for Takiff algebras

Molev

2020

Preprint

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For every simple Lie algebra g we consider the associated Takiff algebra g ℓ defined as the truncated polynomial current Lie algebra with coefficients in g. We use a matrix presentation of g ℓ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra U(g ℓ ). A similar matrix presentation for the affine Kac-Moody algebra g ℓ is then used to prove an analogue of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal-Sugawara vectors for the Lie algebra g ℓ .

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Section: Introductionmentioning

confidence: 81%

Casimir elements and Sugawara operators for Takiff algebras

Molev

2020

Preprint

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“…In other words, g y ⋉ k 14 = q ⋉ q ab . By [PY,Example 4.1], Theorem 2.2 of the same paper [PY] applies to q and guarantees us that the symmetric invariants of q⋉q ab form a polynomial ring in 6 generators, where the degrees of the basic invariants are the same as in the case of q ⊕ q.…”

Section: Proposition 43 (mentioning

confidence: 99%

“…These polynomials are tri-homogeneous w.r.t. the decomposition [PY,Theorem 2.2] applies to q and assures that the symmetric invariants of q⋉q ab form a polynomial ring in 8 generators, where the degrees of the basic invariants are the same as in the case of q ⊕ q.…”

Section: Corollary 45 (Item 5a) the Reductionmentioning

confidence: 99%

Semi-Direct Products Involving Sp2n or Spinn with Free Algebras of Symmetric Invariants

Panyushev

Yakimova

2019

Representations and Nilpotent Orbits of Lie Algebraic Systems

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“…Since the latter work is a crucial ingredient in our paper, we will briefly describe their main results in Section 2.2. More recently, Macedo and Savage extended their work further to the case of truncated multicurrents [MS19], and Panyushev and Yakimova showed that for complex Lie 2 M. Chaffe and L. Topley algebras, the operation g ↦ g 1 preserves the property of having polynomial symmetric invariants, under mild assumptions [PY20].…”

Section: Introductionmentioning

confidence: 99%

Category for truncated current Lie algebras

Chaffe,

Topley

2023

Can. J. Math.-J. Can. Math.

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In this paper, we study an analogue of the Bernstein–Gelfand–Gelfand category ${\mathcal {O}}$ for truncated current Lie algebras $\mathfrak {g}_n$ attached to a complex semisimple Lie algebra. This category admits Verma modules and simple modules, each parametrized by the dual space of the truncated currents on a choice of Cartan subalgebra in $\mathfrak {g}$ . Our main result describes an inductive procedure for computing composition multiplicities of simples inside Vermas for $\mathfrak {g}_n$ , in terms of similar composition multiplicities for ${\mathfrak {l}}_{n-1}$ where ${\mathfrak {l}}$ is a Levi subalgebra. As a consequence, these numbers are expressed as integral linear combinations of Kazhdan–Lusztig polynomials evaluated at 1. This generalizes recent work of the first author, where the case $n=1$ was treated.

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Takiff Algebras With Polynomial Rings of Symmetric Invariants

Cited by 8 publications

References 27 publications

Casimir elements and Sugawara operators for Takiff algebras

Casimir elements and Sugawara operators for Takiff algebras

Semi-Direct Products Involving Sp2n or Spinn with Free Algebras of Symmetric Invariants

Category for truncated current Lie algebras

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