Gradings on the algebra of upper triangular matrices and their graded identities

Vincenzo, Onofrio Mario Di; Koshlukov, Plamen; Valenti, A.

doi:10.1016/j.jalgebra.2003.08.004

Cited by 64 publications

(57 citation statements)

References 18 publications

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“…Hence U = CC s by Lemma 2.4 and this implies that • is equivalent to the symplectic involution s. 2 We have the following easy to apply criterion for determining which class of equivalent involutions some • belongs to. Let (A, * ) be an algebra with involution (of first kind).…”

Section: Proofmentioning

confidence: 99%

“…Here the polynomial in (1) is S 2 -standard; in that in (2), the variables of the monomial z i 1 · · · z i r are not necessarily ordered. In the first case we move the variables from the monomial z i 1 · · · z i r to the first commutator, using the identity (v).…”

Section: But Identity (1) Of Remark 54 Implies Thatmentioning

confidence: 99%

“…Let f be a polynomial of type (2). In case there is only one variable z in f = z i 1 · · · z i r [x j 1 , .…”

Section: But Identity (1) Of Remark 54 Implies Thatmentioning

confidence: 99%

“…The gradings and the graded identities of UT n (F ) have been extensively studied, see for example [2,10] and their references. All gradings on UT n (F ) by a finite abelian group when F is algebraically closed field and char(F ) = 0, were described in [15].…”

Section: Introductionmentioning

confidence: 99%

“…We have for some nonzero polynomial α. Note that the polynomials z 1 · · ·ẑ j · · · z n [y, z j ] and z 1 · · ·ẑ i · · ·ẑ j · · · z n z i [y, z j ] have zero coefficient of e 23 in D since the matrices corresponding to the variables z have zero entry in position (2,2), and [y, z j ] has zero entries on the main diagonal. For the same reasons [y, z j ]z 1 · · ·ẑ j · · · z n has zero coefficient for e 12 .…”

mentioning

confidence: 99%

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Involutions for upper triangular matrix algebras

Vincenzo

Koshlukov

Scala

2006

Advances in Applied Mathematics

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In this paper we describe completely the involutions of the first kind of the algebra UT_n(F ) of n × n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F ) when n is even and a single class in the odd case. Furthermore we consider the algebra UT_2(F ) of the 2 × 2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution ∗, we describe the ∗-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras. Then we consider the ∗-polynomial identities for the algebra UT_3(F ) over a field of characteristic zero. We describe a finite generating set of the ideal of ∗-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the ∗-identities for the algebra U_Tn(F ) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities

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Section: Proofmentioning

confidence: 99%

Section: But Identity (1) Of Remark 54 Implies Thatmentioning

confidence: 99%

“…Let f be a polynomial of type (2). In case there is only one variable z in f = z i 1 · · · z i r [x j 1 , .…”

Section: But Identity (1) Of Remark 54 Implies Thatmentioning

confidence: 99%