How to define the action of U(G) in this situation?

Question

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 $fg(u⊗v)=fgu⊗v+fu⊗gv+gu⊗fv+u⊗fgv$, right? How to state this fact for $V^{\otimes n}$, i.e. $fg$ acting on $u⊗v$, where $u=⊗_{i=1}^{n-k} u_i$ and $v=\otimes_{i=1}^k v_i$, for each $k=1,...,n−1$? Thanks,

Answer 1

If $V_1,V_2\dots,V_n$ are $U(\mathfrak g)$-modules, then the $U(\mathfrak g)$-module structure on the tensor product $V_1\otimes V_2\otimes\cdots\otimes V_n$ is the unique one such that whenever $x\in\mathfrak g\subset U(\mathfrak g)$ we have $$\begin{aligned}g\cdot v_1\otimes\cdots\otimes v_n&=(x\cdot v_1)\otimes v_2\otimes\cdots\otimes v_n\\&+v_1\otimes(x\cdot v_2)\otimes v_3\otimes\cdots\otimes v_n\\{}&+v_1\otimes v_2\otimes(x\cdot v_3)\otimes v_4\otimes\cdots\otimes v_n\\{}&+\cdots\\{}&+v_1\otimes v_2\otimes\cdots\otimes v_{n-1}\otimes(x\cdot v_n).\end{aligned}$$ Notice that this formula is only valid for elements of $\mathfrak g$, not all elements of $U(\mathfrak g)$.

If you want to know how an arbitrary element of $U(\mathfrak g)$ acts on the tensor product, write it as a sum of products of elements of $\mathfrak g$ and make use of the relation $$xy\cdot v = x\cdot(y\cdot v)$$ valid for all $x$, $y\in U(\mathfrak g)$ and all $v\in V_1\otimes V_2\otimes\cdots\otimes V_n$.

You should really pick up a book on Hopf algebras. My favorite is Sweedler's Hopf Algebras.