On the factorability of the ideal of ⁎-graded polynomial identities of minimal varieties of PI ⁎-superalgebras

2022 ◽  
Vol 589 ◽  
pp. 273-286
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Viviane Ribeiro Tomaz da Silva ◽  
Ernesto Spinelli
Keyword(s):  
Polynomial Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

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.

Robinson–Schensted–Knuth Correspondence and Weak Polynomial Identities of M1,1(E)

2005 ◽  
Vol 12 (02) ◽  
pp. 333-349
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Roberto La Scala
Keyword(s):  
Hilbert Series ◽  
Quotient Algebra ◽  
Infinite Field ◽  
Polynomial Identities ◽  
The Ideal

In this paper, it is proved that the ideal Iw of the weak polynomial identities of the superalgebra M1,1(E) is generated by the proper polynomials [x1, x2, x3] and [x2, x1] [x3, x1] [x4, x1]. This is proved for any infinite field F of characteristic different from 2. Precisely, if B is the subalgebra of the proper polynomials of F<X>, we determine a basis and the dimension of any multihomogeneous component of the quotient algebra B / (B ∩ Iw). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson–Schensted–Knuth correspondence defined for single semistandard tableaux of double shape.

scholarly journals On H-simple not necessarily associative algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950162
Author(s):  
A. S. Gordienko
Keyword(s):  
Associative Algebra ◽  
Hopf Algebras ◽  
Simple Algebra ◽  
Polynomial Identities ◽  
Associative Algebras ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Graded Polynomial Identities ◽  
Group Gradings ◽  
Polynomial Formula

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

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