• Title/Summary/Keyword: symmetric algebras

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THE CLASSIFICATION OF ω-LEFT-SYMMETRIC ALGEBRAS IN LOW DIMENSIONS

  • Zhiqi Chen;Yang Wu
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.747-762
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    • 2023
  • ω-left-symmetric algebras contain left-symmetric algebras as a subclass and the commutator defines an ω-Lie algebra. In this paper, we classify ω-left-symmetric algebras in dimension 3 up to an isomorphism based on the classification of ω-Lie algebras and the technique of Lie algebras.

ON THE INDEX AND BIDERIVATIONS OF SIMPLE MALCEV ALGEBRAS

  • Yahya, Abdelaziz Ben;Boulmane, Said
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.385-397
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    • 2022
  • Let (M, [ , ]) be a finite dimensional Malcev algebra over an algebraically closed field 𝔽 of characteristic 0. We first prove that, (M, [ , ]) (with [M, M] ≠ 0) is simple if and only if ind(M) = 1 (i.e., M admits a unique (up to a scalar multiple) invariant scalar product). Further, we characterize the form of skew-symmetric biderivations on simple Malcev algebras. In particular, we prove that the simple seven dimensional non-Lie Malcev algebra has no nontrivial skew-symmetric biderivation.

DISCRETE DUALITY FOR TSH-ALGEBRAS

  • Figallo, Aldo Victorio;Pelaitay, Gustavo;Sanza, Claudia
    • Communications of the Korean Mathematical Society
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    • v.27 no.1
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    • pp.47-56
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    • 2012
  • In this article, we continue the study of tense symmetric Heyting algebras (or TSH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for TSH-algebras bearing in mind the results indicated by Or lowska and Rewitzky in [E. Orlowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no. 1-3, 275-295] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has TSH-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.

ON 2-GENERATING INDEX OF FINITE DIMENSIONAL LEFT-SYMMETRIC ALGEBRAS

  • Yang, Xiaomei;Zhu, Fuhai
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1537-1556
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    • 2017
  • In this paper, we introduce the notion of generating index ${\mathcal{I}}_1(A)$ (2-generating index ${\mathcal{I}}_2(A)$, resp.) of a left-symmetric algebra A, which is the maximum of the dimensions of the subalgebras generated by any element (any two elements, resp.). We give a classification of left-symmetric algebras with ${\mathcal{I}}_1(A)=1$ and ${\mathcal{I}}_2(A)=2$, 3 resp., and show that all such algebras can be constructed by linear and bilinear functions. Such algebras can be regarded as a generalization of those relating to the integrable (generalized) Burgers equation.

Inverse of Frobenius Graphs and Flexibility

  • Aljouiee, Abdulla
    • Kyungpook Mathematical Journal
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    • v.45 no.4
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    • pp.561-570
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    • 2005
  • Weak Crossed Product Algebras correspond to certain graphs called lower subtractive graphs. The properties of such algebras can be obtained by studying this kind of graphs ([4], [5]). In [1], the author showed that a weak crossed product is Frobenius and its restricted subalgebra is symmetric if and only if its associated graph has a unique maximal vertex. A special construction of these graphs came naturally and was known as standard lower subtractive graph. It was a deep question that when such a special graph possesses unique maximal vertex? This work is to answer the question partially and to give a particular characterization for such graphs at which the corresponding algebras are isomorphic. A graph that follows the mentioned characterization is called flexible. Flexibility is to some extend a generalization of the so-called Coxeter groups and its weak Bruhat ordering.

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On the Decomposition of Cyclic G-Brauer's Centralizer Algebras

  • Vidhya, Annamalai;Tamilselvi, Annamalai
    • Kyungpook Mathematical Journal
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    • v.62 no.1
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    • pp.1-28
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    • 2022
  • In this paper, we define the G-Brauer algebras $D^G_f(x)$, where G is a cyclic group, called cyclic G-Brauer algebras, as the linear span of r-signed 1-factors and the generalized m, k signed partial 1-factors is to analyse the multiplication of basis elements in the quotient $^{\rightarrow}_{I_f}^G(x,2k)$. Also, we define certain symmetric matrices $^{\rightarrow}_T_{m,k}^{[\lambda]}(x)$ whose entries are indexed by generalized m, k signed partial 1-factor. We analyse the irreducible representations of $D^G_f(x)$ by determining the quotient $^{\rightarrow}_{I_f}^G(x,2k)$ of $D^G_f(x)$ by its radical. We also find the eigenvalues and eigenspaces of $^{\rightarrow}_T_{m,k}^{[\lambda]}(x)$ for some values of m and k using the representation theory of the generalised symmetric group. The matrices $T_{m,k}^{[\lambda]}(x)$ whose entries are indexed by generalised m, k signed partial 1-factors, which helps in determining the non semisimplicity of these cyclic G-Brauer algebras $D^G_f(x)$, where G = ℤr.

A NOTE ON THE RANK 2 SYMMETRIC HYPERBOLIC KAC-MOODY ALGEBRAS

  • Kim, Yeon-Ok
    • The Pure and Applied Mathematics
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    • v.17 no.1
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    • pp.107-113
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    • 2010
  • In this paper, we study the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We give the sufficient conditions for existence of imaginary roots of square length -2k ($k\;{\in}\;\mathbb{Z}$>0). We also give several relations between the roots on g(A).