MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The usual action of $fg$ on $u⊗v$ , where $f,g$ are 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 $fg(u⊗v)=fgu⊗v+fu⊗gv+gu⊗fv+u⊗fgv$, using the comultiplication, right? How to state this fact for $V^{\otimes n}$, i.e. $fg$ acting on $u⊗v$, where $u=\otimes_{i=1}^{n-k} u_i$ and $v=\otimes_{i=1}^k v_i$, for each $k=1,...,n-1$ ? Thanks,

share|cite|improve this question
The formula you've written is how fg acts on u⊗v. This is just the associativity of tensor product. Honestly, I don't think this is a very appropriate question for MO (you might get a better response on Actually, I think you would probably be better served by just going and reading a book about Hopf algebras, as all these points will be addressed in any reasonable one. – Ben Webster Oct 25 '10 at 0:52
@Ben Webster: I have tried many books. But I haven't got any answer about this specific situation. I found in the Jantzen book that it is possible, but not how to do. Anyway, thanks for the link! – Chris Oct 25 '10 at 1:12
I would also suggest that you ask this on – Mariano Suárez-Alvarez Oct 25 '10 at 2:56
This is, I take it, the question you meant to ask at ? – Theo Johnson-Freyd Oct 25 '10 at 5:35
Chris, your formula only works when $f$ and $g$ are primitive elements. – S. Carnahan Oct 25 '10 at 5:52
up vote 3 down vote accepted

There is nothing deep here. The coproduct $\Delta : U(\mathfrak g) \to U(\mathfrak g)\otimes U(\mathfrak g)$ simply implements Leibniz's product rule: if $v_1\in V_1$ and $v_2\in V_2$, then $x\in \mathfrak g$ acts on $V_1\otimes V_2$ by $x: v_1\otimes v_2 \otimes (xv_1)\otimes v_2 + v_1\otimes (xv_2)$. Extending this, on a large tensor product we have: $$ x(v_1 \otimes v_2 \otimes \cdots \otimes v_n) = xv_1 \otimes v_2 \otimes \cdots \otimes v_n + v_1 \otimes xv_2 \otimes \cdots \otimes v_n + \cdots + v_1 \otimes v_2 \otimes \cdots \otimes xv_n$$ If $x,y\in \mathfrak g$, then $xy \in U(\mathfrak g)$ acts via the composition $x\circ y$. For example, $$ xy \left( \bigotimes_{k=1}^n v_k\right) = \sum_{k=1}^n (xy\text{ acts on }v_k) + \sum_{j\neq k}(x\text{ acts on }v_j\text{ and }y\text{ acts on }v_k)$$

Note that in general, for $f\in U(\mathfrak g)$, we do not have $f(u\otimes v) = fu \otimes v + u\otimes fv$. For example, the constant $1$ is an element of $U(\mathfrak g)$, and $1(u\otimes v) = u\otimes v \neq 2u\otimes v = 1u\otimes v + u\otimes 1v$. More generally, not all differential operators are derivations: certainly you do not expect $\frac{\partial^2}{\partial x^2}$ to satisfy a Leibniz rule!

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.