Questions tagged [quantum-groups]
Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
192 questions with no upvoted or accepted answers
2
votes
0
answers
90
views
Co-quasitriangular Hopf algebra - notation
In one article I found the following statement :
If $A$ is a Hopf algebra over $k$ with co-quasitriangular structure $r$, then one can define map $r_{21}*r:A\otimes A\rightarrow k \ \ $ (...).
...
- 797
2
votes
0
answers
134
views
Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$
I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167:
Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra $...
- 465
2
votes
0
answers
141
views
Quantum Algebras -- Crystal Basis/Graph
Suppose I have a finite-dimensional irreducible $U_q(sl_2)$-module say $V$, and (L,B) is its crystal basis.
How do you find the crystal basis of the evaluation $U'$-module $V_{x=1}$? And is there a ...
2
votes
0
answers
169
views
Outer automorphism for $U_q(\mathfrak{su}(2|2))$
It is known that Lie superalgebra $\mathfrak{su}(2|2)$ (and only this one, not arbitrary $\mathfrak{su}(n|n)$) has the nontrivial central extension which forms an $\mathfrak{sl}_2$ triplet, let's call ...
- 765
1
vote
0
answers
54
views
Coproduct on $U_q(sl_2)$
Recall that $U_q(sl_2)$ is the quotient of the free associative $\mathbb{Q}(q)$-algebra on generators $E$, $F$, $K^{\pm 1}$ such that $KE = q^2 EK$, $KF = q^{-2} FK$, and $[E,F] = \frac{K - K^{-1}}{q -...
- 11
1
vote
0
answers
70
views
Affiliating the whole algebra of 'coordinates' with a locally compact quantum group
When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) ...
- 143
1
vote
0
answers
106
views
How does $R \equiv 1\ (\text {mod}\ h)\ $?
Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if
$(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$
$(2)$ $R_{21} = R^{-1},$
$(...
- 41
1
vote
0
answers
65
views
Looking for paper ‘Lyndon bases and the multiplicative formula for R-matrices’ by M.Rosso
Does anyone have a link to the paper ‘Lyndon bases and the multiplicative formula for R-matrices’ by M. Rosso? I cannot find it in Google Scholar or Z-Lib. Thank you!
1
vote
0
answers
122
views
How to make sense of $\mathrm{Mat}_q(n \times n)$? Are there notions of quantum vector space, quantum linear algebra, etc?
Given some algebra $\mathcal{A}$ and $q \in \mathbb{C}$, we say that a matrix $M \in \mathrm{Mat}(n \times n ; \mathcal{A})$ is a quantum matrix in $\mathrm{Mat}_q(n \times n)$ iff the following ...
- 545
1
vote
0
answers
121
views
Lagrangian subcategories of (non-pointed) braided tensor categories
I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...
- 113
1
vote
0
answers
75
views
Problem in understanding Theorem $6.2.9$ from Chari and Pressley
The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
- 199
1
vote
0
answers
166
views
How to understand a definition in KLR algebra in the setting of quantum affine algebras?
I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra:
$$
X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1)
$$
This ...
- 6,201
1
vote
0
answers
137
views
Representation of quantum groups
Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
- 121
1
vote
0
answers
55
views
Proof that for any $u \in L_0$, there exists a $f(q) \in \mathbb{Q}[q]$ for which $f(q)u \in V^{A}$ in global bases theory
Setting
We consider subrings of $\mathbb{Q}(q)$:
$A_0$: localization at $(q)$
$A_{\infty}$: localization at $(q^{-1})$
$A$ = $\mathbb{Q}[q, q^{-1}]$
and free lattices of $\mathbb{Q}(q)$
$L_0$: ...
- 31
1
vote
0
answers
139
views
Submodules of $V\otimes V^*$
Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
- 353
1
vote
0
answers
285
views
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}[...
- 1,813
1
vote
0
answers
88
views
Reference request: Nichols algebras of a braided vector space with a diagonal braiding
Are there some references of the proof of the following result?
Let $(V, c)$ be a braided vector space over a field $k$ with a basis $x_1, \ldots, x_n$, where $c$ is a diagonal braiding given by
\...
- 6,201
1
vote
0
answers
51
views
Is $\mathcal{O}_q(GL_n(k))$ a Nichols algebra?
Let $k$ be a field. The quantized coordinate ring of the group $GL_n(k)$ is defined in Section 3.1 in the paper. Is $\mathcal{O}_q(GL_n(k))$ a Nichols algebra? Thank you very much.
- 6,201
1
vote
0
answers
55
views
How to improve this argument for a restriction of the universal $R$-matrix to $U_{q}^{+}(\mathfrak{g})\otimes U_{q}^{0}(\mathfrak{g})$?
The standard universal $R$-matrix for quantum group algebra $U_{q}\left(\mathfrak{g}\right)$, where $\mathfrak{g}$ is of finite type, is
$$
R_{q}=exp\left(q\sum_{i,j}\left(B^{-1}\right)_{ij}H_{i}\...
- 357
1
vote
0
answers
68
views
Reference for $q^{\eta\sum_{i}t_{\lambda_{i}}\otimes t_{\mu_{i}}}$ part of an expression for universal $R$-matrix of a quantum group algebra
The Wikipedia entry for quantum groups states that a quantum group $U_{q}\left(\mathfrak{g}\right)$ has an infinite formal sum that plays the role of an $R$ matrix, which is the product of two factors,...
- 357
1
vote
0
answers
62
views
Is $T(V)$ a Yetter-Drinfeld module over $T(V^* \otimes V)$?
Let $V$ be a vector space and $V^*$ the dual vector space. Let $T(V)$ be the tensor algebra of $V$. Is there some action $T(V^* \otimes V) \otimes T(V) \to T(V)$ and coaction $T(V) \to T(V^* \otimes V)...
- 6,201
1
vote
0
answers
102
views
An equation about the Lie bialgebra of a Poisson-Lie group
Let $w^{R}: G \rightarrow \mathfrak{g}\otimes \mathfrak{g}$ be the right translate of the Poisson bivector $w$ of $G$ to the identity, and let $\delta : \mathfrak{g}\rightarrow \mathfrak{g} \otimes \...
- 348
1
vote
0
answers
71
views
Low-dimensional classical r-matrices
Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{...
- 6,201
1
vote
0
answers
120
views
Irreducible representations of quantum affine algebras
The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...
- 6,201
1
vote
0
answers
92
views
Deforming the category of representations of the Yangian of a simple Lie algebra?
Since I got a very good answer to my previous question,
217585,
I am asking a sequel by moving up a level.
Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and ...
- 488
1
vote
0
answers
173
views
quantum deformation
The standard quantum groups, say $GL_q(n)$ or $U_q(gl(n))$, depend on the parameter $q$, which in the classical limit tends toward 1. Let $t:=q-1$ be considered as a generic parameter then we can ...
- 21
1
vote
0
answers
115
views
Are the Standard Quantum Groups Coordinate Rings Noetherian?
Are the standard quantum groups $C_q[G]$ Noetherian and if so what is a standard reference?
- 163
1
vote
0
answers
228
views
Clebsch Gordan coefficients of compact quantum groups
Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
- 11
1
vote
0
answers
216
views
polynomial representation of $sl_{2}(k)$
Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
\...
1
vote
0
answers
256
views
On the decomposition of two representations of the Iwahori-Hecke algebra of type A_n
Consider the Iwahori Hecke algebra $H_q(n)$ of the symmetric group $S(n)$, the $\Bbb{Z}[q^{1/2},q^{-1/2}]$ algebra with generators $T_i$, $1 \leq i \leq n-1$, subject to the braid relations and the ...
- 952
0
votes
0
answers
32
views
Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 923
0
votes
0
answers
92
views
Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
0
votes
0
answers
124
views
Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
0
votes
0
answers
105
views
Concrete examples of quantum duality principle
Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
- 255
0
votes
0
answers
105
views
A variant of quantum harmonic oscillators
We have the following variant of harmonic oscillators.
$$
\left\{
\begin{array}{**lr**}
T = a + a^\dagger\\
a | n \rangle = \sqrt{[n]} |n-1 \rangle \\
a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
- 105
0
votes
0
answers
138
views
Is $[n]_q!$ invertible in $\mathbb C [[h]]\ $?
Consider the Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F, H$ and relations $:$
$$[H, E] = 2 E,\ \ [H, F] = - 2 F,\ \ [E, F] = \frac {q^H - q^{-H}} {q - q^{-...
- 41
0
votes
0
answers
99
views
How to show that quantum $sl_2 (\mathbb C)$ is a Hopf algebra deformation of $U (sl_2 (\mathbb C))\ $?
The quantum $sl_2 (\mathbb C)$ is the non-commutative, non-cocommutative Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F$ and $H$ with the relations $:$
$$[H, ...
- 41
0
votes
0
answers
70
views
Why is $\Pi_r^L$ a 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-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$...
- 199
0
votes
0
answers
70
views
Associativity of Quantum Double
Here is the statement about the associativity of the quantum double of bialgebras in Klimyk-Schmudgen "Quantum Groups ..." (Sec 8.2.1)
Can anyone help me derive the formula of on bottom of ...
- 2,390
0
votes
0
answers
106
views
Hopf algebra antipodes and right left comodule equivalences
Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
- 1,144
0
votes
0
answers
148
views
Reference request: the formula $\langle x, [f, g] \rangle = \langle \delta(x), f \otimes g \rangle$
Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}^*$ the dual vector space of $\mathfrak{g}$ which is also a Lie algebra with natural brackets. Let $\delta: U(\mathfrak{g}) \to U(\mathfrak{g}) \...
- 6,201
0
votes
0
answers
195
views
$h$-adic Completion of $U_q(\frak{sl}_2)$?
Consider the algebra (I am for forgetting the Hopf structure) $U_q(\frak{sl}_2)$ defined over ${\mathbb C}$, and the formal power series version/ $h$-adic version $U_h(\frak{sl}_2)$, which I think as ...
- 1,776