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Results tagged with hopf-algebras
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user 1306
A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.
10
votes
Accepted
Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra?
Do you know the paper of Loday and Pirashvili? They discuss what, in their opinion, should replace the notion of a Hopf algebra in Leibniz setting, "Hopf algebras in the category of linear maps".
- 16.9k
9
votes
Is $Tor_A(k,k)$ a bicommutative Hopf algebra?
This is not true. Consider the algebra $A=T(V)/V^{\otimes 2}$, it is a commutative algebra whose augmentation ideal has zero multiplication. We have $\mathrm{Tor}_A(k,k)\cong T(V[1])$ with the shuffle …
- 16.9k
8
votes
How to recognize a Hopf algebra?
A homological condition that might be useful: in the Hopf case, the Yoneda algebra $Ext_A^\bullet(k,k)$ embeds into the Hochschild cohomology $HH^\bullet(A,A)$, moreover, there is a Gerstenhaber algeb …
- 16.9k
4
votes
On the isomorphism problem of enveloping algebras
FWIW, a ten year old article states: "We stress that, in spite of all this, the
characteristic zero case of the isomorphism problem remains entirely open."
(https://link.springer.com/article/10.1007/ …
- 16.9k