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Results tagged with hopf-algebras
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user 58211
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 $G$.
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About the integral form of a quantum group
As far as I understood, in order to specialize a quantum group $U_q(\mathfrak{g})$, lets say over $\mathbb{Q}(q)$, to an element $\epsilon \in \mathbb{C}^\times$, it is necessary to find a $\mathbb{Z} …
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Examples of basic coalgebras
For an algebraically closed field $k$, let $C$ be a $k$-coalgebra. Given a minimal injective cogenerator $E$, there is a so-called basic coalgebra $B_C=coend^C(E)$, s.t. the comodule categories $Mod^C …
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Categorical interpretation of quantum double $D(A,B,\eta)$
It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties:
The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ca …
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Hopf monoid from comonoidal structures
Let $\mathcal{V}$ be a closed braided monoidal category and $\mathcal{V}-Cat$ the monoidal bicategory of small $\mathcal{V}$-enriched categories. Let $\mathcal{C}$ be a pseudo-comonoid in $\mathcal{V} …
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Orthogonal idempotents with sum equal to 1 in $k[G]$ span sub-Hopf algebra
Let $G$ be a finite group. Let $B$ be a set of orthogonal non-zero idempotents with $|B| \leq |G|$, s.t. $\sum_{b \in B}b =1_{kG}$. Is it known if $B$ spans a sub-Hopf algebra $kH \subseteq kG$?
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Is the associated grouplike $\gamma=uS(u)^{-1}$ of a quasi-triangular Hopf algebra always th...
Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for $H$ fin. dim. pointed with $G(H)$ abelian). Let …
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Tensor algebras in the bicategory $\mathsf{2Vect}$
To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both de …
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Lusztig's definition of quantum groups
In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form $U^{\mathbb{Q}(q)}_q$ that is rather different from the usual approach (see [1,Ch.9.1] for expam …
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