Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

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Abstract

Let A be a superalgebra with graded involution or superinvolution ∗ and let cn∗(A), n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ2 and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519–545 2017)
Lingua originaleEnglish
pagine (da-a)961-976
Numero di pagine16
RivistaAlgebras and Representation Theory
Volume22
Stato di pubblicazionePublished - 2019

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Polynomial Growth
Superalgebra
Involution
Codimension
Multilinear Algebra
Upper triangular matrix
Polynomial Identities
Exponential Growth
Group Algebra
Subalgebra
Classify
If and only if
Algebra

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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title = "Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions",
abstract = "Let A be a superalgebra with graded involution or superinvolution ∗ and let cn∗(A), n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ2 and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519–545 2017)",
author = "{La Mattina}, Daniela and Antonino Giambruno and Antonio Ioppolo",
year = "2019",
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pages = "961--976",
journal = "Algebras and Representation Theory",
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TY - JOUR

T1 - Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

AU - La Mattina, Daniela

AU - Giambruno, Antonino

AU - Ioppolo, Antonio

PY - 2019

Y1 - 2019

N2 - Let A be a superalgebra with graded involution or superinvolution ∗ and let cn∗(A), n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ2 and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519–545 2017)

AB - Let A be a superalgebra with graded involution or superinvolution ∗ and let cn∗(A), n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ2 and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519–545 2017)

UR - http://hdl.handle.net/10447/371736

UR - https://link.springer.com/article/10.1007/s10468-018-9807-3

M3 - Article

VL - 22

SP - 961

EP - 976

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

SN - 1386-923X

ER -