Questions tagged [quantum-groups]
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.
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Does the antipode leave Peter-Weyl blocks invariant.
Let $\mathcal U_q(\mathfrak{g})$ be a algebraic quantum group over $\mathbf{Q}(q)$, and $A_q(G)$ be its linear dual of matrix entires. Suppose I have a matrix entry $c_{f,v}\in L^r(\lambda)\otimes L(\...
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2
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Restricting and extending *-homomorphisms with norm-dense subalgebras
Let $G$ and $H$ be discrete groups, and suppose that I have a surjective *-homomorphism:
$$\pi:\mathrm{C}^*(G)\to C_{\text{r}}(H).$$
Is it true that the restriction to the group algebra $\pi_{|_{\...
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Properties of Bicrossed Product multiplcation
I am reading Kassel's Quantum Groups book (the chapter on Drinfeld doubles). In it, there is the following claim:
If $H,K\subseteq G$ are groups such that $\forall g\in G$, $\exists!(y,z)\in H\times K$...
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Haar integral of a finite dimensional Hopf algebra: an explicit expression
Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
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What do Kassel, Rosso, and Turaev mean by "duality"?
In their book "Quantum Groups and Knot Invariants", Kassel, Rosso, and Turaev prove that $U_q\mathfrak{sl}(N+1)$ has a PBW basis. I'm having trouble following the last step, though.
In ...
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Is the preimage of a Hopf subalgebra a Hopf subalgebra?
The following must be simple, but I have no intuition here, so excuse me.
Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
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"The" Universal R Matrix in Quantum Groups
For a quantum group (a quasitriangular Hopf algebra) $A$, it has a distinguished element in $A \otimes A$ called the universal R matrix in many texts (e.g. Kassel). This confuses me, because nowhere ...
- 185
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Verifying Yang Baxter Equation
$V$ be a 3 dimensional space with basis $\{e_1,e_2,e_3\}$ over $\mathbb{C}$. Let $q\in \mathbb{C}^\ast$.
$\beta: V\otimes V \to V\otimes V$ defined by
\begin{equation}
\beta(e_i\otimes e_j) =
\...
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How to show that $v$ is invertible in $H\ $?
Let $(H, \mu, \eta, \Delta, \varepsilon, S)$ be a Hopf algebra. Let $\mathcal F \in H \otimes H$ be an invertible element such that
$(1)$ $(\mathcal F \otimes 1) (\Delta \otimes \text {id}) (\mathcal ...
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If $H$ is a Hopf algebra deformation of $U (\mathfrak g)$ then $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra.
Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \...
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Relation satisfied in quantum enveloping algebra
I'm currently reading "Quantum Groups", Christian KASSEL, 1995
In chapter XVIII. section 8, he is proving the second rigidity theorem for quantum enveloping algebras, namely if $ A = (U(\...
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Reference request: unusual expansion of product of binomial coefficients
There is a well-known formula for the product of the binomial coefficients:
$$\binom{n}{a}\binom{n}{b}=\sum_{i=0}^{min(a,b)}\binom{a+b-i}{i,a-i,b-i}\binom{n}{a+b-i}$$
I'm interested in a different ...
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How to calculate $\text{End}(V^{\otimes n})$
Let $\mathfrak g$ be a complex semisimple Lie algebra, and $V$ the fundamental $\mathfrak g$-module. Then we can decompose $V^{\otimes n}$ into the direct sum of irreducibles.
For example, in the case ...
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Are all module categories over the category $\mathrm{Rep}_k(G)$ given by $\mathrm{Rep}_k(H)$ for a subgroup $H\subset G$?
I am reading this article, where they classify the semisimple module categories with finitely many irreducibles over the category of representations of quantum $SL(2)$.
They say that, in the classical ...
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Compute $S(E^i F^j K^l)$ in $U_q$
Here is the question I am trying to solve:
Compute $S(E^i F^j K^l)$ in $U_q.$
Here is my thoughts:
Definition: We define $U_q = U_q(\mathfrak{sl}(2))$ as the algebra generated by the four variables $E ...
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2
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Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$
Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$ (as defined in the previous exercise; assume that the field $k$ is of characteristic zero.)
Here is the exercise before it:
Let $L$ ...
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Determining the grouplike elements of a Hopf algebra
Here is the question I am trying to solve:
For any Lie algebra $L$ determine the group of grouplike elements of the Hopf algebra $U(L).$
Some definitions:
1-To any Lie Algebra $L$ we assign an (...
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Proving that a symmetric bilinear form on $L$ is invariant
Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$
...
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Why are the maps $\eta, \mu, \Delta, \varepsilon$ linear?
Here is the question I am trying to solve:
(Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \...
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Proving the uniqueness of a map
Here is the question I am trying to solve:
(Tensor product of coalgebras) Let $(C, \Delta, \varepsilon)$ and $(C', \Delta ', \varepsilon ')$ be coalgebras. Show that the linear maps $\pi: C \otimes C' ...
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What is the meaning of $(\mu \otimes \mu) ({\rm id}\otimes \tau \otimes {\rm id})$?
Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra:
$\require{AMScd}$
$$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\...
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Do multi-parameter unitary subgroups exist?
I'm working with an $N$-dimensional quantum system that is defined by the following Hamiltonian
$$
H = H_{\text{drift}} + \sum_{j=1}^n a^j H_{\text{drive}}^j
$$
Where $n \ll N$ (in my case $n = 4$ ...
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q analogue of a number is a polynomial in $[2]_q$
$[m]_q = \frac{q^m−q^{-m}}{q-q^{-1}}$ is the q- analogue of the number $m\in\mathbb{Z}_{\geq0}$.
$[0]_q=0$, $[1]_q=1$ and $[2]_q=q+q^{-1}$.
I don't know how to prove
Every $[m]_q$ can be expressed as ...
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Why we need an orthonormal basis?
Here is the question I am trying to solve:
Prove that $\lambda$ is injective.
Here is the definition of the linear map $\lambda$:
Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their ...
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What happens if $(f_1\otimes g_1)(u_1 \otimes v_1) = (f_2 \otimes g_2)(u_2 \otimes v_2)$?
Prove that $\lambda$ is injective.
Here is the definition of the linear map $\lambda$:
Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their tensor product $f \otimes g: U \otimes V \to ...
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Poincare Series of a graded algebra (revisited)
Here is the question I am trying to solve:
Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
- 3,635
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Poincare series of a graded algebra
Here is the question I am trying to solve:
Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
- 3,635
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Finding expression for $([[e_r^{\ast},e_k], e_s^{\ast}], e_l).$
Let $(\mathfrak {g}, \mathfrak {g}_+, \mathfrak {g}_-)$ be a finite dimensional Manin triple i.e. $\mathfrak g$ is a finite dimensional Lie algebra endowed with a non-degenerate invariant bilinear ...
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Any non-degenerate triangular structure on $\mathfrak {g}$ gives rise to a left-invariant non-degenerate Poisson structure on $G.$
Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenerate triangular structure on $\mathfrak {g}$ i.e. $...
- 2,802
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When is a triangular structure called non-degenerate?
Let $(\mathfrak {g}, [\cdot, \cdot], \delta)$ be a triangular Lie bialgebra with classical $r$-matrix $r \in \bigwedge^2 \mathfrak {g}.$ In such a case we say that $r$ is a triangular structure on $\...
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Does no one know where the canonical commutation relations come from?
It's been quite a long time, reading about the origins of the mathematical formulation of quantum mechanics, absolutely no one gives a justified derivation of the canonical commutation relations, even ...
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Any symplectic manifold is a Poisson manifold.
Let $M$ be a symplectic manifold with a non-degenerate closed $2$-form $\omega.$ Then for any $f \in C^{\infty} (M)$ and for any vector field $u \in \mathfrak {X} (M)$ there exists a vector field $V_{...
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Quantum plane is a bialgebra
I am reading ‘Hopf algebras and their actions on rings’. Susan wrote the quantum plane as an example at 1.3.9 Example. He said $B = k \langle x,y \mid xy = qyx \rangle$, $0 \neq q \in k$ with ...
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Where does the unitarity structure of $U_q(\mathfrak{sl}_2)$ come from?
It is known that when $q$ is the root of unity, the representation of the quantum group $U_q(\mathfrak{sl}_2)$ is a unitary modular tensor category. However, if we want it to have the dagger structure,...
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References about Quantum Theory
I’m interested in Quantum Theory. Can anyone recommend a good book (for mathematicians) that tackles varios topics (not necessarily formally).
I have in mind books that have the same idea of books ...
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Finding the expression for $f_h^{-1} (a)$ when the expression for $f_h (a)$ is given.
I am following the section $6.1$ on Deformations of Hopf Algebras (Chapter $6$) from A Guide to Quantum Groups written by Chari and Pressley. Let $A$ be a Hopf algebra over $k$ with two deformations $...
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Explicit algebra for elliptic quantum group associated with $SU(2)$, motivated by that of the $q$-deformation
I am not familiar with the mathematical literature on various algebras, but I've briefly come across $q$-deformations a couple times in my work.
The main example I've seen has been the $q$-deformation ...
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``Quantum''/ non-commutating extension of polytopes
Are there non-commutative (ie. quantum) extensions of polytopes?
More specifically I was wondering if there are some deformation, say $\hbar$, to polytopes which when is taken to be zero, one gets ...
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How do I prove this expression for the q-deformed factorial?
Kauffman states without proof in "Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds" that these 2 expressions for the q-deformed factorial are equal:
$$ [n]! = \prod_{k=1}^n \...
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How to construct irreducible representations of a Hopf algebra based on a finite non-Abelian group (beyond quantum double)?
I'm ultimately trying to construct a Hopf algebra based on a finite non-abelian group $G$ such that its irreducible representations:
are labeled by $g\in G$ and $\rho\in\text{irr}(G)$
have the fusion ...
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Is this a valid q-deformation of $SU(2)$ via the $SU(3)$ Lie algebra?
In my physics research, I stumbled upon a rather simple deformation of the $SU(2)$ algebra, and I was wondering whether it qualifies as a `$q$-deformed Lie algebra' (a notion which is unfamiliar to me)...
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How to show that the inversion map is a anti-Poisson map?
Let $G$ be a Poisson-Lie group and the inversion map $i:G\to G$ defined by
$$
g\mapsto g^{-1}.
$$
How to show that the inversion map is anti-Poisson, i.e., $\{f\circ i, g\circ i\}(x) = -\{f, g\}(x^{-1}...
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Minimal tensor product of $B(H)$ and $C(G)$
Let $H$ be a finite dimensional vector space, and $G$ be a compact group. Let $B(H)$ be the bounded operators on $H$, let $C(G)$ be the complex valued continuous functions on $G$, and let $C(G;B(H))$ ...
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Are certain Type-A crystal bases actually "famous" graphs in graph theory?
Is there a well-known name for the triangular "half" of the (square) grid graph (see picture below)?
This is actually a special case of my full question: I am studying the crystal base for ...
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Separating in even and odd powers if they don't commute
During a lesson on Rabi's oscillations, my professor computed the following series:
$$H = \sum_n \left( (a |e_1 \rangle \langle e_0| + a^\dagger |e_0 \rangle \langle e_1|\right)^n$$
separating in odd ...
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Is the multiplier algebra $M(K(H)\otimes C(X))$ isomorphic to $M(K(H))\otimes C(X)$?
In the survey article by Maes and Van Daele in the beginning of section 5, after Def. 5.1, the following claim is made (I will use different notation here, but wanted to give the reference nonetheless)...
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Decomposition of VN(G) into direct sum of matrix algebras when G is a compact group. [closed]
Does anyone know any concrete proof of the following statement which is from the accepted answer to this question? Or any resource that has the proof?
When $G$ is a compact group, by using classical ...
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Anticommutation of Convolution Products on Trace Class Operators of Quantum Groups
Edit: This question has been posted to MathOverflow.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
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Does quasi-triangularity imply invertibility of the antipode?
I have been studying quantum groups from Christian Kassel's Quantum Groups side-by-side with Majid's Foundations of Quantum Group Theory.
I noticed that while Majid proves that the antipode of a quasi-...
- 2,271
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Self-duality of $U_q(b_+)$ proof (Majid, A Quantum Groups Primer, Proposition 2.5)
I am reading A Quantum Groups Primer by Shahn Majid, and I'm having trouble filling in the details for the proof of Proposition 2.5, which states that the Hopf algebra $U_q(b_+)$ is self-dual.
$
\...
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