The algebra of 2 × 2 upper triangular matrices as a commutative algebra: gradings, graded polynomial identities and Specht property

2021 ◽  
Author(s):  
Pedro Morais ◽  
Manuela da Silva Souza
Keyword(s):  
Commutative Algebra ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Graded Polynomial Identities ◽  
Specht Property

GRADED IDENTITIES FOR THE ALGEBRA OF n×n UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD

2003 ◽  
Vol 13 (05) ◽  
pp. 517-526 ◽  
Author(s):  
PLAMEN KOSHLUKOV ◽  
ANGELA VALENTI
Keyword(s):  
Infinite Field ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

We consider the algebra Un(K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤn-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra.

ALGEBRAS SATISFYING THE POLYNOMIAL IDENTITY [x1,x2][x3,x4,x5]=0

2004 ◽  
Vol 03 (02) ◽  
pp. 121-142 ◽  
Author(s):  
ONOFRIO M. DI VINCENZO ◽  
VESSELIN DRENSKY ◽  
VINCENZO NARDOZZA
Keyword(s):  
Polynomial Identity ◽  
Characteristic Zero ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Proper Subvariety

Let [Formula: see text] be a field of characteristic zero, and [Formula: see text] the variety of associative unitary algebras defined by the polynomial identity [x1,x2][x3,x4,x5]=0. This variety is one of the several minimal varieties of exponent 3 (and all proper subvarieties are of exponents 1 and 2). We describe asymptotically its proper subvarieties. More precisely, we define certain algebras ℛ2k for any k∈ℕ and show that if [Formula: see text] is a proper subvariety of [Formula: see text], then the T-ideal of its polynomial identities is asymptotically equivalent to the T-ideal of the identities of one of the algebras [Formula: see text], E, ℛ2k or ℛ2k⊕E, for a suitable k∈ℕ. We give also another description relating the T-ideals of the proper subvarieties of [Formula: see text] with the polynomial identities of upper triangular matrices of a suitable size.

scholarly journals COCHARACTERS OF POLYNOMIAL IDENTITIES OF UPPER TRIANGULAR MATRICES

2012 ◽  
Vol 11 (01) ◽  
pp. 1250018 ◽  
Author(s):  
SILVIA BOUMOVA ◽  
VESSELIN DRENSKY
Keyword(s):  
Explicit Form ◽  
Generating Function ◽  
Characteristic Zero ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices

Let T(Uk) be the T-ideal of the polynomial identities of the algebra of k × k upper triangular matrices over a field of characteristic zero. We give an easy algorithm which calculates the generating function of the cocharacter sequence χn(Uk) = Σλ⊢n mλ(Uk)χλ of the T-ideal T(Uk). Applying this algorithm we have found the explicit form of the multiplicities mλ(Uk) in two cases: (i) for the "largest" partitions λ = (λ1,…,λn) which satisfy λk+1 +⋯+ λn = k - 1; (ii) for the first several k and any λ.

scholarly journals Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

2021 ◽  
Author(s):  
Onofrio M. Di Vincenzo ◽  
Vincenzo Nardozza
Keyword(s):  
Lie Algebra ◽  
Differential Polynomial ◽  
Two Dimensional ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Generating Set ◽  
Upper Triangular Matrices

AbstractWe study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

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