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