Timeline for How to work with co-multiplication?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4, 2010 at 16:22 | vote | accept | Binai | ||
| Feb 28, 2011 at 17:07 | |||||
| Oct 25, 2010 at 16:15 | vote | accept | Binai | ||
| Oct 25, 2010 at 16:15 | |||||
| Oct 25, 2010 at 7:47 | comment | added | Andrew Stacey | And in a bialgebra then the comultiplication is an algebra homomorphism so one would have $\Delta(f g) = \Delta(f) \Delta(g)$ and so on for the iterates. | |
| Oct 24, 2010 at 19:00 | comment | added | Theo Johnson-Freyd | @Chris: The numbering convention is for $\Delta^n$ to denote the map $C \to C^{\otimes n}$ --- it has precisely $n$ outputs. You can extend this backwards: $\Delta^2 = \Delta$, $\Delta^1 = \operatorname{id}$, and, if your coalgebra is counital, $\Delta^0$ is the counit. | |
| Oct 24, 2010 at 17:59 | comment | added | Sean Tilson | if you look at your original definition chris you will see that $\Delta^2 = \Delta$. | |
| Oct 24, 2010 at 17:55 | comment | added | Binai | One more point. Observe that the usual action of $fg$ on $u\otimes v$, where $f,g$ are two elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by $fg (u \otimes v)=fgu \otimes v + fu \otimes gv + gu\otimes fv + u\otimes fgv$ using the comultiplication, right? How to state this fact for $V^{\otimes n}$, i.e. $fg$ acting on $u \otimes v$, where $u=\otimes_{i=1}^{n-k} u_i$ and $v=\otimes_{i=1}^k v_i$? Thanks, | |
| Oct 24, 2010 at 17:34 | comment | added | Binai | Both equations are not consistent! Anyway, have you meant $\Delta^2$ in the first formula or the power is wrong in the second one? | |
| Oct 24, 2010 at 14:43 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
added 207 characters in body
|
| Oct 24, 2010 at 14:35 | history | answered | Mariano Suárez-Álvarez | CC BY-SA 2.5 |