Mixing
Understanding homomorphism from coalgebra to algebra
0
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Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which preserves operations and their neutral elements, but which operations would it preserve between coalgebra and algebra? Thank you.
hopf-algebras
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
$endgroup$
add a comment |
0
$begingroup$
Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which preserves operations and their neutral elements, but which operations would it preserve between coalgebra and algebra? Thank you.
hopf-algebras
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
$endgroup$
$begingroup$
Usually this is done if the domain and range are bialgebras, so they have both a product and a coproduct.
$endgroup$
– Matt Samuel
Jan 6 at 15:46
$begingroup$
Maybe linear map from coalgebra to algebra makes more sense, but I still don't understand how it can be defined. Found it there: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/SeligHopf.pdf , on page 5, section 4 (in the beginning).
$endgroup$
– Ordev Agens
Jan 7 at 18:31
1
$begingroup$
Well it defines it explicitly with a formula. And it's exactly as I said: it's a map between bialgebras (specifically here a bialgebra with itself). In terms of linear maps, you can define a nontrivial linear map between any two nontrivial vector spaces over the same field. It's not a homomorphism, it's just a composition of particular important functions here, including the product and the coproduct.
$endgroup$
– Matt Samuel
Jan 7 at 19:22
$begingroup$
Thank you for explaining. So a linear map from coalgebra ($A$, $mu$, $nu$) to algebra ($A$, $Delta$, $epsilon$) is just a linear map $Ato A$?
$endgroup$
– Ordev Agens
Jan 7 at 19:47
1
$begingroup$
Yes it is. Comment too short.
$endgroup$
– Matt Samuel
Jan 7 at 20:36
add a comment |
0
0
0
$begingroup$
Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which preserves operations and their neutral elements, but which operations would it preserve between coalgebra and algebra? Thank you.
hopf-algebras
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
$endgroup$
Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which preserves operations and their neutral elements, but which operations would it preserve between coalgebra and algebra? Thank you.
hopf-algebras
hopf-algebras
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
asked Jan 6 at 13:02
Ordev AgensOrdev Agens
205
205
$begingroup$
Usually this is done if the domain and range are bialgebras, so they have both a product and a coproduct.
$endgroup$
– Matt Samuel
Jan 6 at 15:46
$begingroup$
Maybe linear map from coalgebra to algebra makes more sense, but I still don't understand how it can be defined. Found it there: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/SeligHopf.pdf , on page 5, section 4 (in the beginning).
$endgroup$
– Ordev Agens
Jan 7 at 18:31
1
$begingroup$
Well it defines it explicitly with a formula. And it's exactly as I said: it's a map between bialgebras (specifically here a bialgebra with itself). In terms of linear maps, you can define a nontrivial linear map between any two nontrivial vector spaces over the same field. It's not a homomorphism, it's just a composition of particular important functions here, including the product and the coproduct.
$endgroup$
– Matt Samuel
Jan 7 at 19:22
$begingroup$
Thank you for explaining. So a linear map from coalgebra ($A$, $mu$, $nu$) to algebra ($A$, $Delta$, $epsilon$) is just a linear map $Ato A$?
$endgroup$
– Ordev Agens
Jan 7 at 19:47
1
$begingroup$
Yes it is. Comment too short.
$endgroup$
– Matt Samuel
Jan 7 at 20:36
add a comment |
$begingroup$
Usually this is done if the domain and range are bialgebras, so they have both a product and a coproduct.
$endgroup$
– Matt Samuel
Jan 6 at 15:46
$begingroup$
Maybe linear map from coalgebra to algebra makes more sense, but I still don't understand how it can be defined. Found it there: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/SeligHopf.pdf , on page 5, section 4 (in the beginning).
$endgroup$
– Ordev Agens
Jan 7 at 18:31
1
$begingroup$
Well it defines it explicitly with a formula. And it's exactly as I said: it's a map between bialgebras (specifically here a bialgebra with itself). In terms of linear maps, you can define a nontrivial linear map between any two nontrivial vector spaces over the same field. It's not a homomorphism, it's just a composition of particular important functions here, including the product and the coproduct.
$endgroup$
– Matt Samuel
Jan 7 at 19:22
$begingroup$
Thank you for explaining. So a linear map from coalgebra ($A$, $mu$, $nu$) to algebra ($A$, $Delta$, $epsilon$) is just a linear map $Ato A$?
$endgroup$
– Ordev Agens
Jan 7 at 19:47
1
$begingroup$
Yes it is. Comment too short.
$endgroup$
– Matt Samuel
Jan 7 at 20:36
$begingroup$
Usually this is done if the domain and range are bialgebras, so they have both a product and a coproduct.
$endgroup$
– Matt Samuel
Jan 6 at 15:46
$begingroup$
Usually this is done if the domain and range are bialgebras, so they have both a product and a coproduct.
$endgroup$
– Matt Samuel
Jan 6 at 15:46
$begingroup$
Maybe linear map from coalgebra to algebra makes more sense, but I still don't understand how it can be defined. Found it there: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/SeligHopf.pdf , on page 5, section 4 (in the beginning).
$endgroup$
– Ordev Agens
Jan 7 at 18:31
$begingroup$
Maybe linear map from coalgebra to algebra makes more sense, but I still don't understand how it can be defined. Found it there: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/SeligHopf.pdf , on page 5, section 4 (in the beginning).
$endgroup$
– Ordev Agens
Jan 7 at 18:31
1
1
$begingroup$
Well it defines it explicitly with a formula. And it's exactly as I said: it's a map between bialgebras (specifically here a bialgebra with itself). In terms of linear maps, you can define a nontrivial linear map between any two nontrivial vector spaces over the same field. It's not a homomorphism, it's just a composition of particular important functions here, including the product and the coproduct.
$endgroup$
– Matt Samuel
Jan 7 at 19:22
$begingroup$
Well it defines it explicitly with a formula. And it's exactly as I said: it's a map between bialgebras (specifically here a bialgebra with itself). In terms of linear maps, you can define a nontrivial linear map between any two nontrivial vector spaces over the same field. It's not a homomorphism, it's just a composition of particular important functions here, including the product and the coproduct.
$endgroup$
– Matt Samuel
Jan 7 at 19:22
$begingroup$
Thank you for explaining. So a linear map from coalgebra ($A$, $mu$, $nu$) to algebra ($A$, $Delta$, $epsilon$) is just a linear map $Ato A$?
$endgroup$
– Ordev Agens
Jan 7 at 19:47
$begingroup$
Thank you for explaining. So a linear map from coalgebra ($A$, $mu$, $nu$) to algebra ($A$, $Delta$, $epsilon$) is just a linear map $Ato A$?
$endgroup$
– Ordev Agens
Jan 7 at 19:47
1
1
$begingroup$
Yes it is. Comment too short.
$endgroup$
– Matt Samuel
Jan 7 at 20:36
$begingroup$
Yes it is. Comment too short.
$endgroup$
– Matt Samuel
Jan 7 at 20:36
add a comment |
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$begingroup$
Usually this is done if the domain and range are bialgebras, so they have both a product and a coproduct.
$endgroup$
– Matt Samuel
Jan 6 at 15:46
$begingroup$
Maybe linear map from coalgebra to algebra makes more sense, but I still don't understand how it can be defined. Found it there: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/SeligHopf.pdf , on page 5, section 4 (in the beginning).
$endgroup$
– Ordev Agens
Jan 7 at 18:31
1
$begingroup$
Well it defines it explicitly with a formula. And it's exactly as I said: it's a map between bialgebras (specifically here a bialgebra with itself). In terms of linear maps, you can define a nontrivial linear map between any two nontrivial vector spaces over the same field. It's not a homomorphism, it's just a composition of particular important functions here, including the product and the coproduct.
$endgroup$
– Matt Samuel
Jan 7 at 19:22
$begingroup$
Thank you for explaining. So a linear map from coalgebra ($A$, $mu$, $nu$) to algebra ($A$, $Delta$, $epsilon$) is just a linear map $Ato A$?
$endgroup$
– Ordev Agens
Jan 7 at 19:47
1
$begingroup$
Yes it is. Comment too short.
$endgroup$
– Matt Samuel
Jan 7 at 20:36