222 research outputs found
Classifying G-graded algebras of exponent two
No full textLet F be a field of characteristic zero and let V be a variety of associative F-algebras graded by a finite abelian group G. If V satisfies an ordinary non-trivial identity, then the sequence cnG(V) of G-codimensions is exponentially bounded. In [2, 3, 8], the authors captured such exponential growth by proving that the limit G(V)=limn→∞cnG(V)nexists and it is an integer, called the G-exponent of V. The purpose of this paper is to characterize the varieties of G-graded algebras of exponent greater than 2. As a consequence, we find a characterization for the varieties with exponent equal to 2
Superinvolutions on upper-triangular matrix algebras
No full textLet UTn(F) be the algebra of nÃn upper-triangular matrices over an algebraically closed field F of characteristic zero. In [18], the authors described all abelian G-gradings on UTn(F) by showing that any G-grading on this algebra is an elementary grading. In this paper, we shall consider the algebra UTn(F) endowed with an elementary Z2-grading. In this way, it has a structure of superalgebra and our goal is to completely describe the superinvolutions which can be defined on it. To this end, we shall prove that the superinvolutions and the graded involutions (i.e., involutions preserving the grading) on UTn(F) are strictly related through the so-called superautomorphisms of this algebra. We shall show that there exist two classes of inequivalent superinvolutions when n is even and a single class otherwise. Along the way, we shall give a complete description of the polynomial identities and the cocharacter sequences of UT2(F) and UT3(F) endowed with all possible superinvolutions
Gradings and graded linear maps on algebras
No full textLet A be a superalgebra over a field F of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and K-gradings on A, where K is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra UTn(F) of n x n upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of UTn(F) endowed with a graded or a pseudo automorphism, for any n, and the ideals of identities with super-automorphism in the cases n = 2 and n = 3
Varieties of algebras with pseudoinvolution: Codimensions, cocharacters and colengths
No full textLet A be a finitely generated superalgebra with pseudoinvolution ⁎ over an algebraically closed field F of characteristic zero. In this paper we develop a theory of polynomial identities for this kind of algebras. In particular, we shall consider three sequences that can be attached to Id⁎(A), the T2⁎-ideal of identities of A: the sequence of ⁎-codimensions cn⁎(A), the sequence of ⁎-cocharacter χ〈n〉⁎(A) and the ⁎-colength sequence ln⁎(A). Our purpose is threefold. First we shall prove that the ⁎-codimension sequence is eventually non-decreasing, i.e., cn⁎(A)≤cn+1⁎(A), for n large enough. Secondly, we study superalgebras with pseudoinvolution having the multiplicities of their ⁎-cocharacter bounded by a constant. Among them, we characterize the ones with multiplicities bounded by 1. Finally, we classify superalgebras with pseudoinvolution A such that ln⁎(A) is bounded by 3. In the last section we relate the ⁎-colengths with the polynomial growth of the ⁎-codimensions: we show that ln⁎(A) is bounded by a constant if and only if cn⁎(A) grows at most polynomially
Vista dal Nord Educazione antimafia e immaginario mafioso in Piemonte e Lombardia
No full text"Vista dal Nord"è il frutto di un lavoro di ricerca sull'immaginario dei giovani studenti lombardi e piemontesi sui fenomeni mafiosi e sull'antimafia, condotto dai Ludovica Ioppolo, Francesca della Ratta-Rinaldi e Giuseppe Ricotta, che già avevano curato degli studi su questo tema tra gli studenti delle scuole superiori in Toscana (2010), nel Lazio (2011), in Liguria e in provincia di Trento (2012). Il lavoro, coordinato da Libera Formazione e dai ricercatori, in collaborazione con i coordinamenti territoriali di Lombardia e Piemonte, è iniziato nel 2013 e nel 2014, con la somministrazione ad un campione di studenti delle scuole superiori di un questionario on-line. Vista dal Nord riporta gli esiti di quanto emerso da questo lavoro, analizzando la grande ricchezza di dati quantitativi e qualitativi emersi, approfondendo in particolare gli effetti delle azioni di educazione antimafia e alla cittadinanza portate avanti da docenti e associazioni nelle scuole italiane."View from the North" is the result of a research work on the imagery of the young students in Lombardy and Piedmont on the mafia and 'antimafia, led by Ludovica Ioppolo, Francesca della Ratta-Rinaldi and Giuseppe Ricotta, who had already taken care of studies on this issue among high school students in Tuscany (2010), in Lazio (2011), in Liguria and in the province of Trento (2012). The work, coordinated by Libera and by the three researchers, in collaboration with the regional coordination of Libera Lombardy and Libera Piedmont, began in 2013 and in 2014, by administering to a sample of high school students of an online questionnaire. View from the North shows the results of the findings of this work, analyzing the great wealth of quantitative and qualitative findings, especially focusing on the effects of anti-Mafia education and citizenship actions carried out by associations and teachers in Italian schools
Standard polynomials and matrices with superinvolutions
No full textLet Mn(F) be the algebra of n x n matrices over a field F of characteristic zero. The superinvolutions ∗ on Mn(F) were classified by Racine in [12]. They are of two types, the transpose and the orthosymplectic superinvolution. This paper is devoted to the study of ∗-polynomial identities satisfied by Mn(F). The goal is twofold. On one hand, we determine the minimal degree of a standard polynomial vanishing on suitable subsets of symmetric or skew-symmetric matrices for both types of superinvolutions. On the other, in case of M2(F), we find generators of the ideal of ∗-identities and we compute the corresponding sequences of cocharacters and codimensions
On multiplicities of cocharacters for algebras with superinvolution
No full textIn this paper we deal with finitely generated superalgebras with superinvolution, satisfying a non-trivial identity, whose multiplicities of the cocharacters are bounded by a constant. Along the way, we prove that the codimension sequence of such algebras is polynomially bounded if and only if their colength sequence is bounded by a constan
Varieties of algebras with pseudoinvolution and polynomial growth
No full textLet A be an associative algebra with pseudoinvolution ∗ over an algebraically closed field of characteristic zero and let cn∗(A) be its sequence of ∗-codimensions. We shall prove that such a sequence is polynomially bounded if and only if the variety generated by A does not contain five explicitly described algebras with pseudoinvolution. As a consequence, we shall classify the varieties of algebras with pseudoinvolution of almost polynomial growth, i.e. varieties of exponential growth such that any proper subvariety has polynomial growth and, along the way, we shall give also the classification of their subvarieties. Finally, we shall describe the algebras with pseudoinvolution whose ∗-codimensions are bounded by a linear function
Graded linear maps on superalgebras
No full textLet A be an associative algebra over a fixed field F of characteristic zero. In this paper we focus our attention on those algebras A graded by Z2, the cyclic group of order 2. In this case A is said to be a superalgebra and it can be decomposed in the direct sum of homogeneous subspaces: A=A0⊕A1. The main goal of this paper is to prove tight relations between some graded linear maps that can be defined on superalgebras, namely involutions, superinvolutions and pseudoinvolutions. Along the way, we shall present a classification of the pseudoinvolutions that one can define on the algebra UTn(F) of n×n upper-triangular matrices. In the final part of the paper we shall also give some consequences of these results in the context of the theory of polynomial identities
Some results concerning the multiplicities of cocharacters of superalgebras with graded involution
No full textLet A be a finitely generated superalgebra with graded involution ⁎ over a field F of characteristic zero and let χnjavax.xml.bind.JAXBElement@1919835,...,njavax.xml.bind.JAXBElement@614d07cf(A), n1+⋯+n4=n, be the (n1,...,n4)-cocharacter of A. In this paper we present some results concerning the multiplicities of cocharacters for this kind of algebras
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