All Questions
Tagged with ra.rings-and-algebras qa.quantum-algebra
78 questions
51
votes
1
answer
2k
views
Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...
- 2,941
28
votes
0
answers
527
views
What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
- 37.8k
18
votes
2
answers
1k
views
Why does Drinfeld Unitarization work?
In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
- 183
16
votes
6
answers
2k
views
What is an algebraic group over a noncommutative ring?
Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I know what "linear ...
- 54.6k
14
votes
1
answer
614
views
Are there interesting semisimple algebras in non-semisimple categories?
Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple ...
- 28.1k
13
votes
1
answer
598
views
Is algebra: ac=ca, bd = db , ad - da = cb - bc ("Manin matrix algebra") - a Koszul algebra?
Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field ...
- 24.8k
12
votes
3
answers
832
views
Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
- 3,318
12
votes
0
answers
605
views
Given an algebra, can it be realized as a block of a Hopf algebra?
During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$...
- 2,450
11
votes
3
answers
1k
views
Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)?
We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra $U(\mathfrak{g})$, my question is: is there any "quantum Lie algebra" embeded into the quantum enveloping ...
- 131
10
votes
1
answer
569
views
Hopf algebra with a non-invertible antipode
What is an example of a Hopf algebra with a non-invertible antipode?
- 145
10
votes
1
answer
518
views
Functoriality of the Hopf dual
Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
- 835
8
votes
1
answer
246
views
commutative "weakly" Frobenius algebras and 2d TQFT
Fix a field $k$. A classic result written up carefully by Abrams in the article "Two-Dimensional Topological Quantum Field Theories and Frobenius Algebras"
says that there is a bijective ...
8
votes
1
answer
1k
views
Hochschild homology of dga's
I'm sure the following statement is well-known to experts:
Let $A$ be a dga. Let $perf(A)$ be the dg-category of perfect dg-modules over A. Then there is a quasi-isomorphism
$$C_\bullet(perf(A)) \to ...
- 460
8
votes
1
answer
248
views
Are annihilation modules in the quantum torus necessarily principal?
I hope that my question yields some standard fact from (noncommutative) ring theory. In discussions with other graduate students, we have outlined some approaches to tackling the question, but haven'...
- 54.6k
8
votes
0
answers
488
views
det(A)det(B) = det(AB+correction), Capelli identities, "factorized" representation of $\mathfrak {gl}_n$
Context: Some probably know that there are Capelli identities which state
$$det(A)det(B) = det(AB+correction)$$ for some matrices with non-commuting elements, they go back to the 19-th century, but ...
- 24.8k
8
votes
0
answers
916
views
duality between universal enveloping and function algebra for GL(n)
Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...
- 5,232
7
votes
2
answers
467
views
Low dimensional noncommutative non-cocommutative Hopf algebras
Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
- 141
7
votes
1
answer
566
views
Is there an integral fusion ring which is not of Frobenius type?
Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...
7
votes
2
answers
386
views
Non-associative deformation quantization
Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
- 3,880
7
votes
1
answer
555
views
Quadratic algebras, quadratic operads, quadratic categories and quantum cohomology
Motivated by the quantisation of the symmetric laws in physics, the category of quadratic algebras has been endowed with two tensor products by Manin in his Montreal lectures notes. These products ...
- 3,652
7
votes
0
answers
385
views
How to define $U_q \mathfrak{g}$ without generators and relations?
I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
- 700
7
votes
0
answers
248
views
Trace on a KLR algebra
The cyclotomic KLR algebra is isomorphic to the Ariki-Koike algebra over a field and so admits a trace (this is used in Hu-Mathas' paper to define bases for the KLR algebra corresponding to Murphy and ...
- 1,413
7
votes
0
answers
460
views
Quantum polynomial rings and singularities
Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with ...
- 3,758
6
votes
2
answers
388
views
monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis
Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e_1,\ldots, e_n$ some basis of $L$. The formula $[e_i,e_j] = \sum_k C_{ij}^k e_k$ determines the structure ...
- 5,232
6
votes
1
answer
231
views
Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?
$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and ...
- 24.8k
6
votes
1
answer
282
views
Quantum exterior algebra
In Generalisation of the quantum exterior algebra the quantum exterior algebra is discussed:
$$
K\langle x_1,\dotsc x_n\rangle/(x_i^2,x_i x_j + q_{i,j}x_j x_i),
$$
with nonzero field elements $q_{i,j}...
- 257
6
votes
2
answers
543
views
Confusion around the reflection equation algebra
I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a ...
- 437
6
votes
1
answer
313
views
What is the smallest rank for a noncommutative fusion ring?
A fusion ring $\mathcal{F}$ (of rank $r$) is given by a finite set $B = \{b_1,b_2, \dots, b_r \}$ such that $b_i b_j = \sum_k n_{i,j}^k b_k$ with $n_{i,j}^k \in \mathbb{Z}_{\ge 0}$, satisfying ...
6
votes
1
answer
272
views
Bialgebraic structure of Sklyanin algebra
Does Sklyanin algebra (which is an elliptic extension of the quantum group) admit a bialgebra structure or even Hopf algebraic structure? Or is it proved that it is impossible to have such a structure?...
- 367
6
votes
1
answer
1k
views
Computing kernels of maps of modules over a finitely presented algebra
I have the following problem: I have an associative (noncommutative) algebra $A$ defined over a rational function field $k = \mathbb{Q}(\delta, \lambda)$. $A$ is given by a presentation in terms of ...
- 2,533
6
votes
0
answers
259
views
Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?
A fusion ring is a finite dimensional $\mathbb{Z}$-module
$\mathbb{Z}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^...
6
votes
0
answers
196
views
Software for BMW algebra calculations?
Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
- 291
6
votes
0
answers
377
views
How to compute the abelianization of the representation theory of a Hopf algebra?
I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...
- 54.6k
5
votes
2
answers
342
views
Classifying Hopf algebras that admit a single irreducible comodule
Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k ...
5
votes
2
answers
280
views
Comodules of Cosemisimple Hopf Algebras
A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
- 1,479
5
votes
2
answers
680
views
Characters on Hopf algebras
For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
5
votes
1
answer
129
views
Covariant splittings of Hopf algebra projections
What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $...
- 1,144
5
votes
2
answers
384
views
a question about finite dimensional representation of a Hopf algebra
Let $H$ be a Hopf algebra over a field $k$ and $V$ a finite
dimensional left $H$-module. Then $End_{k}(V)$ is a right $H$-module
via $(f\cdot h)(v)=S(h_{1})f(h_{2}\cdot v)$.
We set $Ann(End_{k}(V))$={...
5
votes
1
answer
178
views
Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
5
votes
0
answers
345
views
A fusion ring identity
Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
- 309
5
votes
0
answers
219
views
Constructing a noncommutative algebra from a commutative algebra
I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
- 111
4
votes
2
answers
474
views
Does a nonabelian Picard group exist?
Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=...
- 1,143
4
votes
2
answers
615
views
When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?
I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a $...
- 63
4
votes
3
answers
344
views
Coinvariants of tensor products of Hopf algebras
Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...
4
votes
1
answer
193
views
Quotient of quasi-isomorphic cdga's
I'm looking for a theorem about quotient of quasi-isomorphic cdga's:
Let $A, B$ be two cdga's (commutative differential $\mathbb Z$-graded algebra) of nonpositive degrees, and $\mathfrak m \subset A, ...
- 151
4
votes
1
answer
193
views
Bigalois Groupoid Of Drinfel'd Group Double
2-cocycles of a given Hopf algebras $H$ no longer form a group, but a groupoid between different Doi twists of the Hopf algebra $H,L$. The subgroup of "lazy" 2-cocycles precisely preserve the ...
- 1,846
4
votes
1
answer
179
views
quantum affine $gl_2$
There are many sources of the relations and Hopf algebra structure of quantum affine $sl_2$ as a deformed enveloping algebra. However, for an application to integrable systems I need to look at ...
- 1,143
4
votes
1
answer
145
views
Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom?
Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$.
The $n$-th type-A subdivision ...
- 33.8k
4
votes
1
answer
755
views
Commutators for quantum Lie algebras
Can the usual definition of a Lie algebra via commutators be simply adapted
to quantum Lie algebras? Graphically you have the IHX scheme, with the X
being a virtual crossing (so to say). Does it ...
- 4,829
4
votes
0
answers
326
views
Are there infinitely many simple integral fusion rings of rank $4$?
$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...