How to formulate supercommutativity in a characteristic free way?
I believe I've seen an answer to this somewhere (some Deligne notes?) but cannot recover it.
In the $mathbb Z/2$-graded formulation, supercommutativity means that two odd degree elements anticommute while all other kinds of homogeneous elements commute.
If 2 is invertible, to name a $mathbb Z/2$-grading is the same as to name an involution: for $tau:Ato A$ with $tau^2$ equal to identity, even elements are those with $tau e=e$ and the odd ones those with $tau o=-o$. Then every $a$ is uniquely the sum of the even element $(a+tau a)/2$ and the odd element $(a-tau a)/2$.
Now suppose 2 is not invertible, and we still want to formulate supercommutativity in terms of $tau$ alone, in such a way that it becomes precisely supercommutativity as soon as 2 becomes invertible. How to do it?
My attempts produced ugly and inconclusive things like
$$
2yx=xy+xtau(y)+tau(x)y-tau(x)tau(y)
$$
or
$$
yx-tau(y)x-ytau(x)+tau(y)tau(x)=-xy+xtau(y)+tau(x)y-tau(x)tau(y).
$$
There must be something much better, what is it?
superalgebra
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
add a comment |
I believe I've seen an answer to this somewhere (some Deligne notes?) but cannot recover it.
In the $mathbb Z/2$-graded formulation, supercommutativity means that two odd degree elements anticommute while all other kinds of homogeneous elements commute.
If 2 is invertible, to name a $mathbb Z/2$-grading is the same as to name an involution: for $tau:Ato A$ with $tau^2$ equal to identity, even elements are those with $tau e=e$ and the odd ones those with $tau o=-o$. Then every $a$ is uniquely the sum of the even element $(a+tau a)/2$ and the odd element $(a-tau a)/2$.
Now suppose 2 is not invertible, and we still want to formulate supercommutativity in terms of $tau$ alone, in such a way that it becomes precisely supercommutativity as soon as 2 becomes invertible. How to do it?
My attempts produced ugly and inconclusive things like
$$
2yx=xy+xtau(y)+tau(x)y-tau(x)tau(y)
$$
or
$$
yx-tau(y)x-ytau(x)+tau(y)tau(x)=-xy+xtau(y)+tau(x)y-tau(x)tau(y).
$$
There must be something much better, what is it?
superalgebra
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
add a comment |
I believe I've seen an answer to this somewhere (some Deligne notes?) but cannot recover it.
In the $mathbb Z/2$-graded formulation, supercommutativity means that two odd degree elements anticommute while all other kinds of homogeneous elements commute.
If 2 is invertible, to name a $mathbb Z/2$-grading is the same as to name an involution: for $tau:Ato A$ with $tau^2$ equal to identity, even elements are those with $tau e=e$ and the odd ones those with $tau o=-o$. Then every $a$ is uniquely the sum of the even element $(a+tau a)/2$ and the odd element $(a-tau a)/2$.
Now suppose 2 is not invertible, and we still want to formulate supercommutativity in terms of $tau$ alone, in such a way that it becomes precisely supercommutativity as soon as 2 becomes invertible. How to do it?
My attempts produced ugly and inconclusive things like
$$
2yx=xy+xtau(y)+tau(x)y-tau(x)tau(y)
$$
or
$$
yx-tau(y)x-ytau(x)+tau(y)tau(x)=-xy+xtau(y)+tau(x)y-tau(x)tau(y).
$$
There must be something much better, what is it?
superalgebra
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
I believe I've seen an answer to this somewhere (some Deligne notes?) but cannot recover it.
In the $mathbb Z/2$-graded formulation, supercommutativity means that two odd degree elements anticommute while all other kinds of homogeneous elements commute.
If 2 is invertible, to name a $mathbb Z/2$-grading is the same as to name an involution: for $tau:Ato A$ with $tau^2$ equal to identity, even elements are those with $tau e=e$ and the odd ones those with $tau o=-o$. Then every $a$ is uniquely the sum of the even element $(a+tau a)/2$ and the odd element $(a-tau a)/2$.
Now suppose 2 is not invertible, and we still want to formulate supercommutativity in terms of $tau$ alone, in such a way that it becomes precisely supercommutativity as soon as 2 becomes invertible. How to do it?
My attempts produced ugly and inconclusive things like
$$
2yx=xy+xtau(y)+tau(x)y-tau(x)tau(y)
$$
or
$$
yx-tau(y)x-ytau(x)+tau(y)tau(x)=-xy+xtau(y)+tau(x)y-tau(x)tau(y).
$$
There must be something much better, what is it?
superalgebra
superalgebra
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
asked Nov 30 at 9:07
მამუკა ჯიბლაძე
737817
737817
add a comment |
add a comment |
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