Group gradings on the Jordan algebra of upper triangular matrices

Koshlukov, Plamen; Yasumura, Felipe Yukihide

doi:10.1016/j.laa.2017.08.004

Cited by 13 publications

(13 citation statements)

References 11 publications

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“…Here we study the Jordan algebra U T + n of the upper triangular matrices and the gradings on it following [22]. We describe these gradings, and prove that the graded identities they satisfy determine, up to graded isomorphism, the grading.…”

Section: Gradings On U T + Nmentioning

confidence: 99%

Group gradings on the Lie and Jordan algebras of upper triangular matrices

Hitomi

Koshlukov

Yasumura

2017

São Paulo J. Math. Sci.

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Let K be a field and let U T n = U T n (K) denote the associative algebra of upper triangular n × n matrices over K. The vector space of U T n can be given the structure of a Lie and of a Jordan algebra, respectively, by means of the new products: [a, b] = ab − ba, and a • b = ab + ba. We denote the corresponding Lie and Jordan algebra by U T − n and by U T + n , respectively. If G is a group, the G-gradings on U T n were described by Valenti and Zaicev (Arch Math 89(1):33-40, 2007); they proved that each grading on U T n is isomorphic to an elementary grading (that is every matrix unit is homogeneous). Also Di Vincenzo et al. (J Algebra 275(2):550-566, 2004) classified all elementary gradings on U T n. Here we study the gradings and the graded identities on U T − n and on U T +

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Section: Gradings On U T + Nmentioning

confidence: 99%

Group gradings on the Lie and Jordan algebras of upper triangular matrices

Hitomi

Koshlukov

Yasumura

2017

São Paulo J. Math. Sci.

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“…In the non-associative setting, group gradings on the same algebra viewed as a Lie algebra were obtained in [27]. As a Jordan algebra, the classification of group gradings for n = 2 was done in [26], and then, generalized for arbitrary n in [28]. These latter papers concerning the non-associative setting excluded the possibility of characteristic 2 for the base field.…”

Section: Introductionmentioning

confidence: 99%

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

Koshlukov,

Yasumura

2021

Preprint

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We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a complete classification of isomorphism classes of the gradings. We also provide a classification of the practical isomorphism classes of the gradings, which is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is 2, a topic that was neglected in previous works.

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“…Hence, the classifications of abelian group gradings in both cases are equivalent. The Jordan algebra of upper triangular matrices was investigated in [11].…”

Section: Introductionmentioning

confidence: 99%

Group gradings on the Lie and Jordan algebras of block-triangular matrices

Kochetov

Yasumura

2019

Journal of Algebra

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We classify up to isomorphism all gradings by an arbitrary group G on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic 0. It turns out that the support of such a grading always generates an abelian subgroup of G.Assuming that G is abelian, our technique also works to obtain the classification of G-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic 0 and G finite abelian) and classified up to isomorphism by A. Borges et al. in 2018. Finally, still assuming that G is abelian, we classify G-gradings on the upper block-triangular matrices as a Jordan algebra, over an algebraically closed field of characteristic 0. It turns out that, under these assumptions, the Jordan case is equivalent to the Lie case.2010 Mathematics Subject Classification. Primary 17B70, secondary 16W50, 17C99.

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Group gradings on the Jordan algebra of upper triangular matrices

Cited by 13 publications

References 11 publications

Group gradings on the Lie and Jordan algebras of upper triangular matrices

Group gradings on the Lie and Jordan algebras of upper triangular matrices

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

Group gradings on the Lie and Jordan algebras of block-triangular matrices

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