Trace Identities on Diagonal Matrix Algebras

Ioppolo, Antonio; Koshlukov, Plamen; Mattina, Daniela La

doi:10.1007/978-3-030-63111-6_11

Cited by 2 publications

(6 citation statements)

References 20 publications

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“…, α, β = 0, α = β. In [6], the authors presented just some trace identities satisfied by this algebra but no basis of the trace identities was exhibited. Here we are able to address that problem; we prove the following theorem.…”

Section: Trace Identities On Dmentioning

confidence: 99%

“…2 ) holds. In order to obtain the opposite inclusion, first we shall prove that the polynomials (6) Tr…”

Section: Hence the Inclusionmentioning

confidence: 99%

“…We are left to compute the trace codimension sequence. Since the monomials in (6) , where α ∈ F can be non-zero.…”

Section: Hence the Inclusionmentioning

confidence: 99%

“…In this section we focus our attention to the algebra D 2 of 2 × 2 diagonal matrices over the field F . According to Remarks 4, 5, we can define, up to isomorphism, only the following trace functions on D 2 : have been extensively studied in [5,6]. In this section we collect the results concerning their trace T -ideals.…”

Section: Trace Identities On Dmentioning

confidence: 99%

“…Then we prove that the trace identities of D n+1 endowed with a degenerate trace, are very closely related to those of D n with the trace which is in a sense the restriction of the trace of D n+1 to D n . As a consequence, starting from the trace identities of D 2 , determined in [5,6], we find the generators of the trace T -ideal of the identities of D 3 endowed with all possible degenerate traces. Along the way we also compute the trace codimension sequence of these algebras.…”

Section: Introductionmentioning

confidence: 99%

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Matrix algebras with degenerate traces and trace identities

Ioppolo¹,

Koshlukov²,

Mattina³

2020

Preprint

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In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra Dn of n × n diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of D n+1 endowed with a degenerate trace, to those of Dn with the corresponding trace. This allows us to determine the generators of the trace T-ideal of D 3 .In the second part we study commutative subalgebras of M k (F ), denoted by C k of the type F + J that can be endowed with the so-called strange traces: tr(a + j) = αa + βj, for any a + j ∈ C k , α, β ∈ F . Here J is the radical of C k . In case β = 0 such a trace is degenerate, and we study the trace identities satisfied by the algebra C k , for every k ≥ 2. Moreover we prove that these algebras generate the so-called minimal varieties of polynomial growth.In the last part of the paper, devoted to the study of varieties of polynomial growth, we completely classify the subvarieties of the varieties of algebras of almost polynomial growth introduced in ([5]).

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Section: Trace Identities On Dmentioning

confidence: 99%

“…2 ) holds. In order to obtain the opposite inclusion, first we shall prove that the polynomials (6) Tr…”

Section: Hence the Inclusionmentioning

confidence: 99%

“…We are left to compute the trace codimension sequence. Since the monomials in (6) , where α ∈ F can be non-zero.…”

Section: Hence the Inclusionmentioning

confidence: 99%

Section: Trace Identities On Dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Matrix algebras with degenerate traces and trace identities

Ioppolo¹,

Koshlukov²,

Mattina³

2020

Preprint

Self Cite

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Trace codimensions of algebras and their exponential growth

2022

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The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace.Here we study the asymptotic behaviour of the sequence of trace codimensions c tr n (A) and of pure trace codimensions c ptr n (A) of a finite dimensional algebra A over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits limn→∞ n c tr n (A) and limn→∞ n c ptr n (A) always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2.

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Trace Identities on Diagonal Matrix Algebras

Cited by 2 publications

References 20 publications

Matrix algebras with degenerate traces and trace identities

Matrix algebras with degenerate traces and trace identities

Trace codimensions of algebras and their exponential growth

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