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Results tagged with hopf-algebras
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user 11877
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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How to translate multi-segments to Drinfeld polynomials?
This is described in equation (5.2) in the paper which follows from the paper: Quantum affine algebras and affine Hecke algebras.
- 6,201
7
votes
1
answer
181
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How to translate multi-segments to Drinfeld polynomials?
Let $\hat{H}_m=\hat{H}_m(q)$ be the Iwahori-Hecke algebra of $GL_m$, see for example, Section 2. The simple $\hat{H}_m$-modules are parametrized by Zelevinsky's multi-segments, See Section 2.2 of the …
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4
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0
answers
301
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Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie al...
Let $V$ be a finite dimensional vector space. Let $T(V)$ be the tensor algebra over $V$.
Do we have $T(V) \cong S(Lie(V))$ as a graded vector space? Here $S(Lie(V))$ is the symmetric algebra of the …
- 63.9k
3
votes
1
answer
242
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Are there p-adic or finite field analogue of Drinfeld-Jimbo's quantum groups
Drinfeld-Jimbo's quantum groups are associated algebras over the field of complex numbers. Are there some references about the analogue of Drinfeld-Jimbo's quantum groups over a p-adic field or a fini …
- 8,435
3
votes
1
answer
123
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Is $T(V) \rtimes T(V^* \otimes V)$ a bialgebra?
Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$.
The algebras $T(V)$ and $T(V^* \otimes V)$ are bialgebras. I am trying to find some bialgebra stru …
- 1,028
1
vote
1
answer
126
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Action is determined by a braiding
Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by
\begin{align}
& \Psi: V \otimes W \to W \otimes V, \\
& \P …
- 1,028
4
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1
answer
865
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How to show that the graded dual of the universal enveloping algebra of a free Lie algebra o...
In the article, the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X.
It is said that the graded dual of the universal envel …
- 3,738
2
votes
0
answers
310
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Module algebras and comodule algebras
Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-c …
- 63.9k
1
vote
1
answer
199
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Why in the definition of a Nichols algebra we require that $V$ is a Yetter-Drinfeld module?
In the article, a Nichols algebra is defined as follows. Let ${\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}}$. There exists a largest ideal ${\displaystyle {\mathfrak {I}}\subset TV} $ with the follo …
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3
votes
1
answer
682
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Deformation of a Hopf algebra
A deformation of a Hopf algebra is defined as follows.
On page 171 of the book a guide to quantum groups, Remark 2, it is said that Any deformation of a Hopf algebra $A$ as a bialgebra is automatica …
CommunityBot
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1
vote
1
answer
138
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Analog between groups and Hopf algebras
"Subgroups" of a group correspond to "left coideal subalgebras" of a Hopf algebra. Why "subgroups" do not corresponds to "Hopf subalgebras" but "left coideal subalgebras"?
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1
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1
answer
345
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What are all primitive elements in a tensor algebra?
Let $H$ be a Hopf algebra and $V$ a Yetter-Drinfeld module over $H$. Then there is a braiding $\Psi: V \otimes V \to V \otimes V$ given by $\Psi(x \otimes y) = x_{(-1)}.y \otimes x_{(0)}$, where $x_{( …
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How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding
We can also prove that Yetter-Drinfeld condition implies that $\Psi$ is a braiding as follows.
Let $u, v, w \in V $. Then
\begin{align}
& \Psi_{12} \Psi_{23} \Psi_{12} (u \otimes v \otimes w) \\
& = …
- 6,201
3
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2
answers
243
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How to show that Yetter-Drinfeld condition is equialent to $\Psi$ is a braiding
Let $H$ be a Hopf algebra and $V$ a right $H$-module and right $H$-comodule. The module $V$ is a Yetter-Drinfeld module over $H$ if and only if
\begin{align}
( v \triangleleft h_{(2)} )_{(0)} \otimes …
CommunityBot
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2
votes
1
answer
93
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Are braided commutators primitive elements of a braided Hopf algebra?
Let $H$ be a braided Hopf algebra. The multiplication on $H \otimes H$ is defined by $(a \otimes b)(c \otimes d) = a \Psi(b \otimes c) d$, $a,b,c,d \in H$.
Let $H = T(V)$. There is a algebra map $\D …
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