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1. • CommentRowNumber1.
   • CommentAuthorBartek
   • CommentTimeMar 23rd 2017
   • (edited Mar 23rd 2017)
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   Author: Bartek
   Format: MarkdownItexIn the nLab article on the [universal enveloping algebra](https://ncatlab.org/nlab/show/universal+enveloping+algebra#hopf_algebra_structure_on_), the section describing the Hopf algebra structure originally stated that "the coproduct $\Delta: U L \to U(L \coprod L)\cong U L\otimes UL$ is induced by the diagonal map $L \to L \coprod L$." I assume that this is a mistake, and I have since changed the coproduct $\coprod$ to a product $\times$. However, I don't know a great deal about Hopf algebras, so please correct me if I've made a mistake here.

   In the nLab article on the universal enveloping algebra, the section describing the Hopf algebra structure originally stated that “the coproduct $\Delta: U L \to U(L \coprod L)\cong U L\otimes UL$ is induced by the diagonal map $L \to L \coprod L$.”

   I assume that this is a mistake, and I have since changed the coproduct $\coprod$ to a product $\times$. However, I don’t know a great deal about Hopf algebras, so please correct me if I’ve made a mistake here.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMar 24th 2017
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   Author: zskoda
   Format: MarkdownItexIn vector spaces finite sums and finite products are the same. Still, I think that viewing it as a direct sum is better here.

   In vector spaces finite sums and finite products are the same. Still, I think that viewing it as a direct sum is better here.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeMar 24th 2017
   • (edited Mar 24th 2017)
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   Author: zskoda
   Format: MarkdownItexIf we consider the enveloping as the quotient of the tensor algebra, recall that the tensor algebra is the infinite direct sum of tensor powers. Then $(V\coprod W)^{\otimes n}\cong \coprod_{m=0}^n \left(V^{\otimes m}\otimes W^{\otimes (n-m)}\right)$ in the classical case. $$ \coprod_n (V\coprod W)^{\otimes n} \cong \coprod_n V^{\otimes n}\otimes \coprod_m W^{\otimes m} $$

   If we consider the enveloping as the quotient of the tensor algebra, recall that the tensor algebra is the infinite direct sum of tensor powers. Then $(V\coprod W)^{\otimes n}\cong \coprod_{m=0}^n \left(V^{\otimes m}\otimes W^{\otimes (n-m)}\right)$ in the classical case.

   $$\coprod_n (V\coprod W)^{\otimes n} \cong \coprod_n V^{\otimes n}\otimes \coprod_m W^{\otimes m}$$