Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra W(a,b)

Liu, Deng; Hong, Yanyong; Zhou, Hao; Zhang, Nuan

doi:10.1080/00927872.2018.1468903

Cited by 6 publications

(4 citation statements)

References 23 publications

(73 reference statements)

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“…And the extended annihilation algebra is Lie W(a, b, r) e = C∂ Lie W(a, b, r) + , which satisfies (14) and…”

Section: Definition 22 a Conformal Module M Over A Lie Conformal Almentioning

confidence: 99%

“…If r = 0, from (14), by a straightforward calculation, one can show that (3) holds. If a = 0, 1 and r = 0, by (14), one can immediately get […”

Section: Definition 22 a Conformal Module M Over A Lie Conformal Almentioning

confidence: 99%

“…Finite irreducible conformal modules over W(b) were classified in [16] and of the extensions in [12]. A more general class of Lie conformal algebras W(a, b), constructed as a semi-direct sum of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one, was introduced in [14]. A complete classification of finite irreducible conformal modules of W(a, b) was recently given in [15].…”

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confidence: 99%

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Classification of finite irreducible conformal modules over Lie conformal algebra <inline-formula><tex-math id="M1">$ \mathcal{W}(a, b, r) $</tex-math></inline-formula>

Liu

Xiao

Yue

2021

era

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<p style='text-indent:20px;'>We study a family of non-simple Lie conformal algebras <inline-formula><tex-math id="M2">$ \mathcal{W}(a,b,r) $</tex-math></inline-formula> (<inline-formula><tex-math id="M3">$ a,b,r\in {\mathbb{C}} $</tex-math></inline-formula>) of rank three with free <inline-formula><tex-math id="M4">$ {\mathbb{C}}[{\partial}] $</tex-math></inline-formula>-basis <inline-formula><tex-math id="M5">$ \{L, W,Y\} $</tex-math></inline-formula> and relations <inline-formula><tex-math id="M6">$ [L_{\lambda} L] = ({\partial}+2{\lambda})L,\ [L_{\lambda} W] = ({\partial}+ a{\lambda} +b)W,\ [L_{\lambda} Y] = ({\partial}+{\lambda})Y,\ [Y_{\lambda} W] = rW $</tex-math></inline-formula> and <inline-formula><tex-math id="M7">$ [Y_{\lambda} Y] = [W_{\lambda} W] = 0 $</tex-math></inline-formula>. In this paper, we investigate the irreducibility of all free nontrivial <inline-formula><tex-math id="M8">$ \mathcal{W}(a,b,r) $</tex-math></inline-formula>-modules of rank one over <inline-formula><tex-math id="M9">$ {\mathbb{C}}[{\partial}] $</tex-math></inline-formula> and classify all finite irreducible conformal modules over <inline-formula><tex-math id="M10">$ \mathcal{W}(a,b,r) $</tex-math></inline-formula>.</p>

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“…And the extended annihilation algebra is Lie W(a, b, r) e = C∂ Lie W(a, b, r) + , which satisfies (14) and…”

Section: Definition 22 a Conformal Module M Over A Lie Conformal Almentioning

confidence: 99%

“…If r = 0, from (14), by a straightforward calculation, one can show that (3) holds. If a = 0, 1 and r = 0, by (14), one can immediately get […”

Section: Definition 22 a Conformal Module M Over A Lie Conformal Almentioning

confidence: 99%

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Classification of finite irreducible conformal modules over Lie conformal algebra <inline-formula><tex-math id="M1">$ \mathcal{W}(a, b, r) $</tex-math></inline-formula>

Liu

Xiao

Yue

2021

era

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“…where b ∈ K. This is the Gel'fand-Dorfman bialgebra corresponding to the Lie conformal algebra W (1, b) studied in [15]. Note that W (1, 0) is just the Heisenberg-Virasoro conformal algebra introduced in [18].…”

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confidence: 99%

Extending structures for Gel'fand-Dorfman bialgebras

Wen¹,

Hong²

2022

Preprint

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Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of special Lie conformal algebras called quadratic Lie conformal algebras. In this paper, we investigate the extending structures problem for Gel'fand-Dorfman bialgebras, which is equivalent to some extending structures problem of quadratic Lie conformal algebras. Explicitly, given a Gel'fand-Dorfman bialgebra (A, •, [•, •]), this problem asks that how to describe and classify all Gel'fand-Dorfman bialgebraic structures on a vector space) is a subalgebra of E up to an isomorphism whose restriction on A is the identity map. Motivated by the theories of extending structures for Lie algebras and Novikov algebras, we construct an object GH 2 (V, A) to answer the extending structures problem by introducing a definition of unified product for Gel'fand-Dorfman bialgebras, where V is a complement of A in E. In particular, we investigate the special case when dim(V ) = 1 in detail.

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Extending Structures for Gel’fand-Dorfman Bialgebras

Wen,

Hong

2023

Acta. Math. Sin.-English Ser.

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Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra W(a,b)

Cited by 6 publications

References 23 publications

Classification of finite irreducible conformal modules over Lie conformal algebra <inline-formula><tex-math id="M1">$ \mathcal{W}(a, b, r) $</tex-math></inline-formula>

Classification of finite irreducible conformal modules over Lie conformal algebra <inline-formula><tex-math id="M1">$ \mathcal{W}(a, b, r) $</tex-math></inline-formula>

Extending structures for Gel'fand-Dorfman bialgebras

Extending Structures for Gel’fand-Dorfman Bialgebras

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