Timeline for Bialgebra pairing on ring polynomial $K[x]$
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| May 5, 2017 at 21:10 | answer | added | Vladimir Dotsenko | timeline score: 4 | |
| May 5, 2017 at 20:52 | comment | added | მამუკა ჯიბლაძე | Your $t$, if exists, is uniquely determined by $t(1,x)=t(x,1)=\varepsilon(x)$, $t(x,x^2)=t(\Delta(x),x\otimes x)$, $t(x^2,x)=t(x\otimes x,\Delta(x))$, $t(x,x^3)=t(\Delta(x),x\otimes x^2)=t(\Delta(x),x^2\otimes x)$, etc. You just have to check whether what you obtain in this way is consistent or not. | |
| May 5, 2017 at 20:42 | history | edited | Ben McKay | CC BY-SA 3.0 |
corrected the notation and the question
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| May 5, 2017 at 20:13 | comment | added | unknown | However, to avoid confusion the primitive property is not needed here. | |
| May 5, 2017 at 20:12 | comment | added | unknown | There are plenty of bilinear maps $t:U\otimes U \to k$ that satisfy $t(x,x)=1$ and are not bialgebra pairings. For example on a basis, $t(x^n,x^m)=1 if m=n=1$, and zero otherwise. The point is that there exists a (unique) pairing $t$ which is a bialgebra pairing and fulfills $t(x,x)=1$. I want to know how to define the pairing. | |
| May 5, 2017 at 20:00 | comment | added | unknown | Sorry it is typo it is all the same field $k$ through out. For example let $U=k[x]$ with the Hopf algebra structure determined by $x \in P(U)$. I mean the space $P(U)$ gives the coalgebra structure on the polynomial ring. | |
| May 5, 2017 at 19:53 | comment | added | Ben McKay | I guess that $K$ is a field, but what is $k$? Is $K$ an extension field of $k$, perhaps? | |
| May 5, 2017 at 19:52 | comment | added | Ben McKay | When you write $x \in P(K[x])$, do you mean two different things by the same letter $x$, one on the right hand side and the other one the left hand side? | |
| May 5, 2017 at 19:42 | comment | added | unknown | &P(K[x])$ is the Space of primitive elements. | |
| May 5, 2017 at 19:04 | comment | added | Ben McKay | What is $P(K[x])$? | |
| May 5, 2017 at 19:03 | history | edited | Ben McKay | CC BY-SA 3.0 |
spelling, grammar, formatting
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| May 5, 2017 at 18:58 | review | First posts | |||
| May 5, 2017 at 19:00 | |||||
| May 5, 2017 at 18:56 | history | asked | unknown | CC BY-SA 3.0 |