Research output: Contribution to Journal/Magazine › Journal article › peer-review

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In: Journal of Algebra, Vol. 318, No. 2, 15.12.2007, p. 933-952.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Levy, P 2007, 'Varieties of modules for Z/2Z×Z/2Z', *Journal of Algebra*, vol. 318, no. 2, pp. 933-952. https://doi.org/10.1016/j.jalgebra.2007.06.024

Levy, P. (2007). Varieties of modules for Z/2Z×Z/2Z. *Journal of Algebra*, *318*(2), 933-952. https://doi.org/10.1016/j.jalgebra.2007.06.024

Levy P. Varieties of modules for Z/2Z×Z/2Z. Journal of Algebra. 2007 Dec 15;318(2):933-952. doi: 10.1016/j.jalgebra.2007.06.024

@article{a47a7de36e2e4c1f870a886ed13317a6,

title = "Varieties of modules for Z/2Z×Z/2Z",

abstract = "Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n×n)-matrices (A,B) such that A2=B2=[A,B]=0, is equidimensional. C can be identified with the {\textquoteleft}variety of n-dimensional modules{\textquoteright} for Z/2Z×Z/2Z, or equivalently, for k[X,Y]/(X2,Y2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e2=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z×Z/2Z-modules.",

keywords = "Lie algebras in positive characteristic",

author = "Paul Levy",

year = "2007",

month = dec,

day = "15",

doi = "10.1016/j.jalgebra.2007.06.024",

language = "English",

volume = "318",

pages = "933--952",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "ELSEVIER ACADEMIC PRESS INC",

number = "2",

}

TY - JOUR

T1 - Varieties of modules for Z/2Z×Z/2Z

AU - Levy, Paul

PY - 2007/12/15

Y1 - 2007/12/15

N2 - Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n×n)-matrices (A,B) such that A2=B2=[A,B]=0, is equidimensional. C can be identified with the ‘variety of n-dimensional modules’ for Z/2Z×Z/2Z, or equivalently, for k[X,Y]/(X2,Y2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e2=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z×Z/2Z-modules.

AB - Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n×n)-matrices (A,B) such that A2=B2=[A,B]=0, is equidimensional. C can be identified with the ‘variety of n-dimensional modules’ for Z/2Z×Z/2Z, or equivalently, for k[X,Y]/(X2,Y2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e2=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z×Z/2Z-modules.

KW - Lie algebras in positive characteristic

U2 - 10.1016/j.jalgebra.2007.06.024

DO - 10.1016/j.jalgebra.2007.06.024

M3 - Journal article

VL - 318

SP - 933

EP - 952

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -