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Results tagged with quantum-groups
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user 805
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.
4
votes
Accepted
Examples of canonical bases
The complete calculation is done in Lusztig's book (Lemma 42.1.2). Essentially, the condition $b\geq a+c$ comes from the Serre relations; e.g. we have
$$E_1E_2E_1=E_1^{(2)}E_2+E_2E_1^{(2)}.$$
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7
votes
Computing in quantum groups
One very effective tool for automated computation in the positive or negative half of a quantum group is the shuffle algebra realization; see Leclerc's paper (and the references therein) for some of t …
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4
votes
Accepted
Motivations of cross product
It's the Hopf-algebraic version of the semidirect product of groups. To see this, just consider the case you have groups $G, H, G\rtimes H$. The group rings (over, say, a field $k$) are Hopf algebras …
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6
votes
Accepted
"Quantum Littlewood-Richardson" Rule?
Yes, there certainly is a quantum version of the Littlewood-Richardson decomposition (in the generic parameter case) for types $A,B,C,D$ (I don't know about the exceptional types). The generalized Lit …
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7
votes
Intuition behind the definition of quantum groups
Here is a possible affirmative answer to question (2). (Which is to say, I am not aware of any non-quantum way to answer this question.) I assume from the original post that it is sufficient motivatio …
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