8
$\begingroup$

I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \otimes A \to A$ is a morphism in $\mathcal{C}$?

Let $H$ be a bialgebra and $V$ a Yetter-Drinfeld module over $H$. Let $YD_{H}^H$ be the monoidal category of all Yetter-Drinfeld modules over $H$. Since $YD_H^H$ is a tensor category, the tensor algebra $T(V)$ is a Yetter-Drinfeld module over $H$. Therefore $T(V)$ is an object in $YD_H^H$. When we want to show that $T(V)$ is an algebra in $YD_H^H$, we need to show that the multiplication $m: T(V)\to T(V) \otimes T(V)$ is a homomorphsim of $H$-modules? That is, $h.(ab)=(h_{(1)}.a)(h_{(2)}.b)$ for $h \in H, a,b \in T(V)$. Is this correct? Thank you very much.

$\endgroup$
8
$\begingroup$

What you are talking about is the notion of monoid in a monoidal category. To show $A$ is a monoid ('algebra'), you need to construct a multiplication map $\mu: A \times A \to A$, that is associative, where $\times$ is the monoidal product for your monoidal category. In an example like a tensor algebra, you already have associativity, so what you need to show is that multiplication is a homomorphism of Yetter-Drinfeld modules.

Information on monoids in monoidal categories can be found on the nlab, or in the massive work of Aguiar and Mahajan on monoidal functors, species, and hopf algebras: http://www.math.cornell.edu/~maguiar/a.pdf.

$\endgroup$
4
$\begingroup$

The answer to the specific situation you describe is as follows: Yes, $T(V)$ is an algebra in Yetter-Drinfeld modules over $H$ (using the concept that Jacob White mentions in his answer, where in addition we also have a $1$, which is a map from the tensor unit to the algebra satisfying the unital property expressed as a commutative diagram). A reference for this fact for $T(V)$ is, for example, [AS, 2.1]. You need to check two things:

  • The multiplication is a map of algebras (you state this condition),
  • The multiplication is a map of coalgebras (this is missing).

In fact, more is true. As $YD_{H}^H$ is a braided monoidal category, we can speak of bialgebra and Hopf algebra objects in it. $T(V)$ is a (graded, braided) Hopf algebra object in this category. This concept is for example defined in [AS, 1.7].

[AS]: Andruskiewitsch, Nicolás; Schneider, Hans-Jürgen. Pointed Hopf algebras. New directions in Hopf algebras, 1--68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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