Ideals in the Enveloping Algebra of the Positive Witt Algebra

AbstractAfter introducing the q-analogue of the enveloping algebra of the Witt algebra, we construct q-analogues of the module of tensor fields over the Witt algebra and prove a partial q-analogue of Kaplansky's Theorem concerning this module of tensor fields.

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Maps from the enveloping algebra of the positive Witt algebra to regular algebras

Pacific Journal of Mathematics ◽

10.2140/pjm.2016.284.475 ◽

2016 ◽

Vol 284 (2) ◽

pp. 475-509 ◽

Cited By ~ 2

Author(s):

Susan Sierra ◽

Chelsea Walton

Keyword(s):

Witt Algebra ◽

Enveloping Algebra ◽

Regular Algebras

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The universal enveloping algebra of the Witt algebra is not noetherian

Advances in Mathematics ◽

10.1016/j.aim.2014.05.007 ◽

2014 ◽

Vol 262 ◽

pp. 239-260 ◽

Cited By ~ 3

Author(s):

Susan J. Sierra ◽

Chelsea Walton

Keyword(s):

Universal Enveloping Algebra ◽

Witt Algebra ◽

Enveloping Algebra

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CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

Transformation Groups ◽

10.1007/s00031-021-09643-2 ◽

2021 ◽

Author(s):

MÁTYÁS DOMOKOS ◽

VESSELIN DRENSKY

Keyword(s):

Lie Algebra ◽

Associative Algebra ◽

Invariant Theory ◽

Reductive Group ◽

Symmetric Tensor ◽

Free Associative Algebra ◽

Tensor Algebra ◽

Enveloping Algebra ◽

Finite Dimensional ◽

Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

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W-representation of Rainbow tensor model

Journal of High Energy Physics ◽

10.1007/jhep05(2021)228 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Bei Kang ◽

Lu-Yao Wang ◽

Ke Wu ◽

Jie Yang ◽

Wei-Zhong Zhao

Keyword(s):

Partition Function ◽

Matrix Model ◽

Crucial Role ◽

Elementary Function ◽

Tensor Model ◽

Witt Algebra ◽

Virasoro Constraints ◽

Compact Expression ◽

Non Gaussian ◽

Dual Expression

Abstract We analyze the rainbow tensor model and present the Virasoro constraints, where the constraint operators obey the Witt algebra and null 3-algebra. We generalize the method of W-representation in matrix model to the rainbow tensor model, where the operators preserving and increasing the grading play a crucial role. It is shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operator increasing the grading. We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model. Furthermore, we discuss the case of the non-Gaussian red tensor model and present a dual expression for partition function through differentiation.

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PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Transformation Groups ◽

10.1007/s00031-021-09650-3 ◽

2021 ◽

Author(s):

DMITRI I. PANYUSHEV ◽

OKSANA S. YAKIMOVA

Keyword(s):

Poisson Bracket ◽

Lie Algebra ◽

Invariant Theory ◽

Cyclic Permutation ◽

Poisson Brackets ◽

Enveloping Algebra ◽

Order Automorphism ◽

Finite Dimensional ◽

Compatible Poisson Brackets ◽

Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

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