Mixing
An identity related to antipode of a Hopf algebra
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Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $sumlimits_{(h)} h_2 otimes S^{-1}(h_1) = sumlimits_{(h)} h_1 otimes S(h_2)$ hold for any $h in H$, where $Delta(h)=sum h_1 otimes h_2$? Thank you.
hopf-algebras
asked Dec 19 '18 at 9:25
19851985
63
$endgroup$
add a comment |
$begingroup$
Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $sumlimits_{(h)} h_2 otimes S^{-1}(h_1) = sumlimits_{(h)} h_1 otimes S(h_2)$ hold for any $h in H$, where $Delta(h)=sum h_1 otimes h_2$? Thank you.
hopf-algebras
asked Dec 19 '18 at 9:25
19851985
63
$endgroup$
$begingroup$
Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort.
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– Vee Hua Zhi
Dec 19 '18 at 9:47
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I could prove the equality for cocommutative Hopf algebras. As in that case $sum h_2 otimes S^{-1}(h_1) =sum h_1 otimes S^{-1}(h_2)=sum h_1 otimes S(h_2)$. But I do not know in general.
$endgroup$
– 1985
Dec 20 '18 at 5:24
add a comment |
$begingroup$
Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $sumlimits_{(h)} h_2 otimes S^{-1}(h_1) = sumlimits_{(h)} h_1 otimes S(h_2)$ hold for any $h in H$, where $Delta(h)=sum h_1 otimes h_2$? Thank you.
hopf-algebras
asked Dec 19 '18 at 9:25
19851985
63
$endgroup$
Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $sumlimits_{(h)} h_2 otimes S^{-1}(h_1) = sumlimits_{(h)} h_1 otimes S(h_2)$ hold for any $h in H$, where $Delta(h)=sum h_1 otimes h_2$? Thank you.
hopf-algebras
hopf-algebras
asked Dec 19 '18 at 9:25
19851985
63
asked Dec 19 '18 at 9:25
19851985
63
asked Dec 19 '18 at 9:25
19851985
63
asked Dec 19 '18 at 9:25
19851985
63
asked Dec 19 '18 at 9:25
19851985
63
63
$begingroup$
Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort.
$endgroup$
– Vee Hua Zhi
Dec 19 '18 at 9:47
$begingroup$
I could prove the equality for cocommutative Hopf algebras. As in that case $sum h_2 otimes S^{-1}(h_1) =sum h_1 otimes S^{-1}(h_2)=sum h_1 otimes S(h_2)$. But I do not know in general.
$endgroup$
– 1985
Dec 20 '18 at 5:24
add a comment |
$begingroup$
Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort.
$endgroup$
– Vee Hua Zhi
Dec 19 '18 at 9:47
$begingroup$
I could prove the equality for cocommutative Hopf algebras. As in that case $sum h_2 otimes S^{-1}(h_1) =sum h_1 otimes S^{-1}(h_2)=sum h_1 otimes S(h_2)$. But I do not know in general.
$endgroup$
– 1985
Dec 20 '18 at 5:24
$begingroup$
Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort.
$endgroup$
– Vee Hua Zhi
Dec 19 '18 at 9:47
$begingroup$
Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort.
$endgroup$
– Vee Hua Zhi
Dec 19 '18 at 9:47
$begingroup$
I could prove the equality for cocommutative Hopf algebras. As in that case $sum h_2 otimes S^{-1}(h_1) =sum h_1 otimes S^{-1}(h_2)=sum h_1 otimes S(h_2)$. But I do not know in general.
$endgroup$
– 1985
Dec 20 '18 at 5:24
$begingroup$
I could prove the equality for cocommutative Hopf algebras. As in that case $sum h_2 otimes S^{-1}(h_1) =sum h_1 otimes S^{-1}(h_2)=sum h_1 otimes S(h_2)$. But I do not know in general.
$endgroup$
– 1985
Dec 20 '18 at 5:24
add a comment |
1 Answer
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Observe that your requirement is equivalent to the following
$$Delta= (S^2otimes mathrm{id})circtaucirc Delta,$$
where $tau$ is the flip map.
For a cocommutative Hopf algebra it is known that $S^2=mathrm{id}$, therefore in that case it is true.
In general, consider the Sweedler's Hopf algebra as an counterexample.
answered Jan 12 at 19:04
mikismikis
1,4381821
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1 Answer
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1 Answer
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active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
Observe that your requirement is equivalent to the following
$$Delta= (S^2otimes mathrm{id})circtaucirc Delta,$$
where $tau$ is the flip map.
For a cocommutative Hopf algebra it is known that $S^2=mathrm{id}$, therefore in that case it is true.
In general, consider the Sweedler's Hopf algebra as an counterexample.
answered Jan 12 at 19:04
mikismikis
1,4381821
$endgroup$
add a comment |
$begingroup$
Observe that your requirement is equivalent to the following
$$Delta= (S^2otimes mathrm{id})circtaucirc Delta,$$
where $tau$ is the flip map.
For a cocommutative Hopf algebra it is known that $S^2=mathrm{id}$, therefore in that case it is true.
In general, consider the Sweedler's Hopf algebra as an counterexample.
answered Jan 12 at 19:04
mikismikis
1,4381821
$endgroup$
add a comment |
$begingroup$
Observe that your requirement is equivalent to the following
$$Delta= (S^2otimes mathrm{id})circtaucirc Delta,$$
where $tau$ is the flip map.
For a cocommutative Hopf algebra it is known that $S^2=mathrm{id}$, therefore in that case it is true.
In general, consider the Sweedler's Hopf algebra as an counterexample.
answered Jan 12 at 19:04
mikismikis
1,4381821
$endgroup$
Observe that your requirement is equivalent to the following
$$Delta= (S^2otimes mathrm{id})circtaucirc Delta,$$
where $tau$ is the flip map.
For a cocommutative Hopf algebra it is known that $S^2=mathrm{id}$, therefore in that case it is true.
In general, consider the Sweedler's Hopf algebra as an counterexample.
answered Jan 12 at 19:04
mikismikis
1,4381821
answered Jan 12 at 19:04
mikismikis
1,4381821
answered Jan 12 at 19:04
mikismikis
1,4381821
answered Jan 12 at 19:04
mikismikis
1,4381821
1,4381821
add a comment |
add a comment |
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Welcome to MSE! It is more likely that you'll receive an answer if you showed us that you've made an effort.
$endgroup$
– Vee Hua Zhi
Dec 19 '18 at 9:47
$begingroup$
I could prove the equality for cocommutative Hopf algebras. As in that case $sum h_2 otimes S^{-1}(h_1) =sum h_1 otimes S^{-1}(h_2)=sum h_1 otimes S(h_2)$. But I do not know in general.
$endgroup$
– 1985
Dec 20 '18 at 5:24