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Results tagged with rt.representation-theory
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user 85967
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
1
vote
Indecomposable representations of euclidean quivers
I am not particularly knowledgeable on the subject but I remember a recent workshop i attended where some lecturer referred to the book Finite Dimensional Algebras and Quantum Groups mentioning that i …
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4
votes
Accepted
Typical and atypical modules for Lie superalgebras
Regarding the "what is happening in the super case"; yes i agree that in some sense, it has to do with the odd simple roots but i think it is deeper than that:
In the case of semisimple, complex, Lie …
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12
votes
Accepted
Semisimple super Lie algebras
Yes there is a complete classification of finite dimensional, simple Lie superalgebras (over $\mathbb{C}$), which -up to a certain extent- goes very much in parallel with the corresponding case of Lie …
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11
votes
Simple Subalgebras of Simple Lie Algebras
I am not sure if this is exactly what you are looking for, but there have been some classic works, developing general methods for such topics:
In Dynkin, Semisimple subalgebras of semisimple Lie al …
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5
votes
Accepted
Character formula for Lie superalgebras
I agree with the suggestion in the comments for searching the front of the math arXiv (as an entry point), because this is a quite broad and active topic (and i am not sure it can be fully covered wit …
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10
votes
Up to date summary on semisimple Hopf algebra over $\mathbb{C}$
This is a question on an active area of research, with lots of work on it (for the general case of algebraically closed fields of char zero). It is historically and conceptually closely connected to K …
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6
votes
Accepted
Representation of Heisenberg-Weyl elements and their exponentials
The Heisenberg-Weyl algebra or the Weyl algebra or the algebra of the Canonical Commutation Relations (CCR) is generated by the $p,q$ generators subject to the relation
$$
[q, p] = i \hbar I \ \ \ \ …
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4
votes
Hopf Subalgebras of Quantized Algebras
Since the OP is asking for examples of
sub-Hopf algebras which are not generated by the standard generators
i.e. the Chevalley generators (which are actually the generators of the Cartan–Weyl b …
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4
votes
Accepted
Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators
Although i have some doubts as to what the OP is exactly looking for (see my comments above), i hope that the following will be of some interest for its purposes. In:
$U_q(sl(n))$ Difference Operator …
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2
votes
Hopf dual of the Hopf dual
Regarding your first question: the answer is generally no, the restricted dual of the restricted dual of $A$ is generally not isomorphic to $A$:
$$
(A^{\circ})^\circ\ncong A
$$
as has already been ind …
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5
votes
Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
Since you mention classification results for $R$-matrices:
For finite abelian groups, there is a bijection between the set of universal $R$-matrices of the group hopf algebra $\mathbb C[G]$, the set o …
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5
votes
2
answers
399
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 whi …
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5
votes
Category of bicomodules of a cosemisimple Hopf algebra
The answer is yes, if we are talking about finite dimensional, Hopf algebras over a field:
$\bullet$ $H$ being cosemisimple (as a coalgebra) is equivalent to the dual hopf algebra $H^*$ being semi …
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1
vote
Representation theory in braided monoidal categories
I will try to provide an answer for a particular case of your last question: Let us consider (following my comment above) the case of $H=\mathbb{CZ}_2$ i.e. the group hopf algebra equipped with its no …
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7
votes
Examples of representations of quantum groups
If i have correctly understood your question, there are various such examples, arising from mathematical physics contexts (where some of the original motivations for the study of quantum groups first …
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