Let F be a field of characteristic 0 and let E be the infinite dimensional Grassmann algebra over F . In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the n × n upper triangular matrix algebra U Tn(E) with entries in E, lying in a strip of a fixed size. In the second part we compute the double Hilbert series H(E; T k , Y l ) of E, then we define the (k, l)multiplicity series of any PI-algebra. As an application, we derive from H(E; T k , Y l ) an easy algorithm determining the (k, l)-multiplicity series of U Tn(E).
Let U TnpF q be the algebra of the n ˆn upper triangular matrices and denote U TnpF q p´q the Lie algebra on the vector space of U TnpF q with respect to the usual bracket (commutator), over an infinite field F . In this paper, we give a positive answer to the Specht property for the ideal of the Zn-graded identities of U TnpF q p´q with the canonical grading when the characteristic p of F is 0 or is larger than n ´1. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of U TnpF q p´q , is finitely based.Moreover we show that if F is an infinite field of characteristic p " 2 then the Z 3 -graded identities of U T p´q 3 pF q do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of U T p´q 3 pF q, and which is not finitely generated as an ideal of graded identities.
Neste trabalho estudaremos a caracterização de variedades minimais de álgebras associativas de posto básico finito sobre um corpo de característica zero com PI-expoente maior ou igual que dois, isto é, vamos provar que toda variedade deste tipo é gerada por uma álgebra de matrizes triangulares superiores em blocos. Além disso, estudaremos seus T-ideais e finalmente mostraremos uma relação entre a dimensão de Gelfand-Kirillov e o PI-expoente das álgebras relativamente livres de posto finito de uma variedade não nilpotente. Esta dissertação está baseada no artigo de Antonio Giambruno e Mikhail Zaicev publicado por Advances in Mathematics em 2003.
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