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2 votes
1 answer
141 views

Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$

Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring. There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
Justin Bloom's user avatar
13 votes
0 answers
332 views

Lie theory for quantum groups?

$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives: Compact quantum groups in the sense of Woronowicz. Deformation of the universal enveloping algebra of a Lie algebra in ...
user82261's user avatar
  • 357
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 ...
Raoul's user avatar
  • 163
0 votes
0 answers
51 views

Action of Hopf algebra of identity supported distributions on a Lie group

The Hopf algebra of identity supported distributions on a lie group is cocommutative. It is well known that it is a group object in the category of cocommutative coalgebras. Is there a canonical ...
Lefevres's user avatar
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 ...
yohei ohta's user avatar
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^{-...
Anacardium's user avatar
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 $(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$ $(2)$ $R_{21} = R^{-1},$ $(...
Anacardium's user avatar
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, ...
Anacardium's user avatar
4 votes
1 answer
175 views

Drinfeld-Jimbo quantum groups for $q=0$

In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? ...
Jake Wetlock's user avatar
  • 1,144
5 votes
1 answer
319 views

Malcev completion of free groups

Let $K$ be a field with $\operatorname{char} K=0$, $\hat{L}_n$ the complete free Lie algebra of $n$ variables $x_1,\dotsc,x_n$ and $\exp(\hat{L}_n)$ its associated group with the product given by BCH ...
Qwert Otto's user avatar
3 votes
0 answers
151 views

Is there a classical version of Yetter-Drinfeld modules?

One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$. If we think ...
Antoine Labelle's user avatar
1 vote
1 answer
136 views

Two (or less) filtrations on coenveloping coalgebra

Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) ...
Denis T's user avatar
  • 4,600
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-...
Anil Bagchi.'s user avatar
2 votes
1 answer
293 views

Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
double-function's user avatar
9 votes
1 answer
332 views

The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$

I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
Quin Appleby's user avatar
2 votes
0 answers
96 views

Primitive elements in Hopf algebras over the integers

Let $H$ be a Hopf algebra over $\mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $\mathbb Z$). Are there nice, natural ...
grok's user avatar
  • 2,519
4 votes
1 answer
101 views

Non-cosemisimple duals of pointed Hopf algebras

I take the following quote from an answer to this question A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. The quantized enveloping algebras and ...
Piet Bongers's user avatar
8 votes
0 answers
229 views

Integral Milnor-Moore theorem

Given a field K of char. zero the theorem of Milnor Moore states that taking the enveloping hopf algebra defines an embedding $\mathcal{U} $ from Lie algebras over K into hopf algebras over K. Taking ...
Hadrian Heine's user avatar
3 votes
0 answers
91 views

Hopf algebras structure and quantum affine algebras

I'm looking for some information about the Hopf algebras structure and the quantum groups. In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
Mar Pao's user avatar
  • 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 ...
cl4y70n____'s user avatar
2 votes
0 answers
70 views

Embedding problems on quantum groups?

We work over the field of complex numbers. We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
user11090426's user avatar
10 votes
3 answers
1k views

Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
Nadia SUSY's user avatar
5 votes
2 answers
403 views

Indecomposable, non-simple, modules of quantum groups at roots of unity

Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ (i.e., $d$ is the smallest positive integer for ...
Konstantinos Kanakoglou's user avatar
5 votes
0 answers
218 views

Lusztig's completion for universal enveloping algebra

In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
userabc's user avatar
  • 677
3 votes
0 answers
122 views

It there a nice way to describe the structure of Malcev-complete groups?

Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
J. Darné's user avatar
  • 273
5 votes
1 answer
578 views

Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative functions

Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
Ender Wiggins's user avatar
5 votes
1 answer
579 views

The Ungraded Milnor-Moore Theorem

Let $k$ be a field of characteristic $0$. There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...
user avatar
3 votes
1 answer
295 views

When is this map of Hopf algebras Surjective?

I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero. Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ ...
user avatar
10 votes
1 answer
191 views

Exceptional Quantum Groups as FRT-Algebras

Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...
Alesandro Levi's user avatar
12 votes
2 answers
624 views

On the isomorphism problem of enveloping algebras

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...
Mathematician 42's user avatar
2 votes
0 answers
163 views

Quantum invariant: The canonical $2$-tensor

In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
Mathematician 42's user avatar
17 votes
2 answers
2k views

Examples of representations of quantum groups

I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
asv's user avatar
  • 21.8k
1 vote
0 answers
201 views

The order of the antipode in a Hopf algebra

As a result of Radford, any finite-dimensional Hopf algebra an antipode of finite order. My question: How can we classify all finite-dimensional Hopf algebras whose antipode is identity? Here are ...
double-function's user avatar
13 votes
2 answers
997 views

Can one define quantized universal enveloping algebras in a basis-free way?

(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...
Gro-Tsen's user avatar
  • 32.5k
5 votes
0 answers
217 views

DGLA related to the deformation of hopf Algebras

Recently I was considering Hopf algebras and Drinfeld's twists. I stumbled upon a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by ...
Niek de Kleijn's user avatar
4 votes
0 answers
302 views

Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie algebra over $V$?

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 ...
Jianrong Li's user avatar
  • 6,201
4 votes
1 answer
900 views

How to show that the graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra

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 ...
Jianrong Li's user avatar
  • 6,201
3 votes
0 answers
406 views

Semisimple Lie algebras and the commutator algebra

Suppose $A$ is a associative unital $k$-algebra, where $\operatorname{char}k=0$. As is well-known, $A$ becomes a Lie algebra with respect to the commutator bracket $[x, y] = xy-yx$ for $x,y \in A$. ...
Paul Gilmartin's user avatar
4 votes
0 answers
626 views

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 ...
Bipolar Minds's user avatar
4 votes
0 answers
226 views

Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$. Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
user56909's user avatar
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\\ \...
2 votes
0 answers
283 views

universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$, $U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$ and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...
sife's user avatar
  • 491
5 votes
0 answers
281 views

Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?

More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
Qiaochu Yuan's user avatar
9 votes
1 answer
437 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
darij grinberg's user avatar
3 votes
2 answers
365 views

Does there exist a canonical "degree" filtration on quantum groups?

For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of $\bigoplus_{k=0}^...
John Pardon's user avatar
  • 18.7k
13 votes
4 answers
5k views

What is the universal enveloping algebra?

Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...
Bugs Bunny's user avatar
  • 12.3k
12 votes
1 answer
840 views

Comparing two similar procedures for quantizing a Casimir Lie algebra

My primary reference for this question is the very good book Quantum Groups and Knot Invariants by C. Kassel, M. Rosso, and V. Turaev. I'm also drawing from P. Etingof and O. Schiffmann, Lectures on ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
298 views

In what way do exact sequences of Lie ideals integrate to the category of groups?

Please excuse, very naive question: Suppose $g$ is a topological Lie algebra over Q and $G$ = $exp(g)$ the associated group (take free group on formal symbols $exp(X)$, X $\in$ $G$, and impose all ...
olli_jvn's user avatar
  • 904
12 votes
2 answers
1k views

Is there a canonical Hopf structure on the center of a universal enveloping algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\...
Theo Johnson-Freyd's user avatar
45 votes
16 answers
8k views

What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series $$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty \frac{(-x)^n}{(n+1)!}...
Theo Johnson-Freyd's user avatar