Jordan-Lie inner ideals of finite dimensional associative algebras

Baranov, Alexander; Shlaka, Hasan M.

doi:10.1016/j.jpaa.2019.07.011

Cited by 9 publications

(4 citation statements)

References 16 publications

(37 reference statements)

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“…In [13], Knus, proved that The map induces a one-to-one correspondence between the equivalence classes of non-degenerate symmetric and skew-symmetric bilinear forms on multiplication modulo by a factor in and involutions (of the first kind) on End Definition (2.23) [3] Let be a subspace of Then is called 1.a Jordan-Lie I-ideal (or simply, -Lie) if is an I-ideal with Proposition (2.25) [3] Suppose that is a simple with involution and . Then every -Lie of is -perfect.…”

Section: Remarkmentioning

confidence: 99%

Inner Ideals of The Symplectic Simple Lie Algebra

Kareem

Shlaka²

2022

J. Phys.: Conf. Ser.

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Let A be an associative algebra over a field 𝔽 of any characteristic with involution and let K = skew (A) = { a ∈ A | a* = − a} be its corresponding sub-algebra under the Lie product [a, b] = ab − ba for all a, b ∈ A. In this paper, inner ideals of such Lie algebras were defined, considered, studied, and classified. Some examples and results were provided. It is proved that for every Jordan-Lie inner ideal of K, one can find an idempotent e ∈ A such that this inner ideal may be written in the form eK e*. It is also proved inner ideals of such Lie algebras are regular.

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Section: Remarkmentioning

confidence: 99%

Inner Ideals of The Symplectic Simple Lie Algebra

Kareem

Shlaka²

2022

J. Phys.: Conf. Ser.

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“…1-Perfect and Conical Local Systems Definition 5.1. [10] Let A be an associative algebra over a field F. Then A is called 1-perfect if A has no ideals of codimension 1.…”

Section: Perfect Local Systemsmentioning

confidence: 99%

“…Lemma 5.2. [10] (i) The sum of 1-perfect ideals of an associative algebra A is a 1perfect ideal of A.…”

Section: Perfect Local Systemsmentioning

confidence: 99%

Local Systems of Simple Locally Finite Associative Algebras

Shlaka¹

2021

Preprint

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“…Lemma 3.12. [8] Let L be a perfect Lie algebra and let B be an L-perfect inner ideal of L. Suppose that L = i∈I L i , where each L i is an ideal of L. Then B = i∈I B i , where…”

Section: Inner Idealsmentioning

confidence: 99%

Generalization of Jordan-Lie of Finite Dimensional Associative Algebras

Shlaka¹

2021

Preprint

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We generalize Baranov and Shlaka's results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let A be a finite dimensional 1-perfect associative algebras A over an algebraically closed field F of characteristic p ≥ 0 and let A ′ be a subalgebra of A. We prove that for any bar-minimal Jordan-Lie inner ideal B ′ a A ′ , there is a bar-minimal Jordan-Lie inner ideal of A that contains B ′ and if B' is regular, then is a regular inner ideal of A that contains B ′ . We also prove that for any strict orthogonal pair (e ′ , f ′ ) in A ′ , there is a strict orthogonal idempotent pair (e, f ) in A such that e ′ A ′ f ′ ⊆ eAf .

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Jordan-Lie inner ideals of finite dimensional associative algebras

Cited by 9 publications

References 16 publications

Inner Ideals of The Symplectic Simple Lie Algebra

Inner Ideals of The Symplectic Simple Lie Algebra

Local Systems of Simple Locally Finite Associative Algebras

Generalization of Jordan-Lie of Finite Dimensional Associative Algebras

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