Polynomial identities for the Jordan algebra of upper triangular matrices of order 2

Koshlukov, Plamen; Martino, Fabrizio

doi:10.1016/j.jpaa.2012.03.009

Cited by 38 publications

(13 citation statements)

References 8 publications

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“…When n = 2, the gradings and the graded identities of U T + 2 were described in [19]; this will be a particular case of our results. Let 1 = e 11 + e 22 , a = e 11 − e 22 , b = e 12 , then a grading on U T + 2 is isomorphic to one of the following: 1.…”

Section: Gradings On U T + Nmentioning

confidence: 60%

“…This, combined with the results from [11], in which the elementary gradings on the upper triangular matrices were described, provides the complete picture for the gradings on the latter algebra. Also in [19] the authors described all gradings on the Jordan algebra of upper triangular matrices of order 2. Recall that this algebra is the algebra of a symmetric bilinear form (though the form is degenerate).…”

Section: Introductionmentioning

confidence: 99%

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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: 60%

Section: Introductionmentioning

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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“…For the particular case n = 2, we recall the description of the gradings on UJ 2 given in [9]. Theorem 1 ([9]).…”

Section: Introductionmentioning

confidence: 99%

“…The above theorem will be a particular case of our results. We recall that in [9] the authors also described the graded polynomial identities satisfied by each of the possible gradings, including the trivial one. Here we are not going to do that.…”

Section: Introductionmentioning

confidence: 99%

Group gradings on the Jordan algebra of upper triangular matrices

Koshlukov

Yasumura

2017

Linear Algebra and its Applications

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Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra UJn of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on UJn are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.

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Polynomial identities for the Jordan algebra of upper triangular matrices of order 2

Cited by 38 publications

References 8 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 Jordan algebra of upper triangular matrices

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