All Questions
Tagged with hopf-algebras lie-algebras
51 questions
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)!}...
- 54.6k
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 ...
- 21.8k
15
votes
3
answers
2k
views
Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra?
A Leibniz algebra L may be thought of as a noncommutative generalisation of a Lie algebra. One drops the requirement that the bracket be alternating and substitutes the Jacobi identity for the ...
- 33.3k
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). ...
- 12.3k
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&...
- 32.5k
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 ...
- 357
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 \...
- 395
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 $(\...
- 54.6k
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 ...
- 54.6k
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 ...
- 835
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 ...
- 459
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 ...
- 33.8k
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 ...
- 141
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 ...
- 1,361
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 ...
- 985
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 ...
- 7,746
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(\...
user30211
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}...
- 718
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 ...
- 677
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 ...
- 397
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 ...
- 118k
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? ...
- 1,144
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
...
- 145
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 ...
- 6,201
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 ...
- 6,201
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 ...
- 1,813
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)$ ...
- 41
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$ ...
user30211
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}^...
- 18.7k
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 ...
- 904
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 ...
- 3,446
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 ...
- 31
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 ...
- 273
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$. ...
- 426
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[...
- 542
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 ...
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 ...
- 2,519
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 ...
- 165
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 ...
- 395
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 ...
- 491
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) ...
- 4,600
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
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
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
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 ...
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\\
\...
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
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 ...
- 35
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
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