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Results tagged with reference-request
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user 703
This tag is used if a reference is needed in a paper or textbook on a specific result.
0
votes
Reference for quantum Schur-Weyl duality
This is treated in section 8.6 of the book "Quantum Groups and Their Representations" by Klimyk and Schmudgen, although also not using the quantum coordinate algebra.
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7
votes
2
answers
487
views
Idempotency of the q-antisymmetrizer
Background
When constructing the exterior algebra of a (finite-dimensional, complex) vector space $V$, there are two equivalent pictures. The first is the quotient picture. First you define the ten …
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10
votes
Lie algebras over non-algebraically closed fields
Well, certainly things get more complicated when the field is not algebraically closed, as you can see from the classification of finite-dimensional simple Lie algebras over $\mathbb{R}$. But there a …
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6
votes
Explicit Computations of Examples in Spin Geometry
Appendix A to Chapter 9 of the book Elements of Noncommutative Geometry by Gracia-Bondia, Varilly, and Figueroa is titled "Spin geometry of the Riemann sphere". It is 15 pages long and goes into quit …
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4
votes
Clifford Lie Algebras
A little bit of what you want can be found in Chapter 5 of Gracia-Bondia, Varilly, and Figueroa's book Elements of Noncommutative Geometry. They don't say much about subalgebras, I think, but they do …
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5
votes
A quantum Grothendieck group?
The forgetful functor from the category of Hopf algebras to the category of bialgebras has a left adjoint. This means that given a bialgebra $B$, there is a Hopf algebra $H(B)$ with a bialgebra morph …
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11
votes
A book for problems in Functional Analysis
I realy like the exercises in Gert Pedersen's book Analysis Now.
5
votes
Accepted
Non-commutative versions of X/G
Noncommutative versions of sheaves and holomorphic functions are not very well understood. Better understood are noncommutative versions of measurable, continuous, or smooth functions. I generally w …
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2
votes
analytic structure on lie groups
I don't know the original reference, but you can find a proof of the theorem about real-analytic structures on Lie groups in Chapter 1 of Knapp's book "Lie Groups Beyond an Introduction." The proof u …
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2
votes
Accepted
Completing The Space Sections in a Vectorbundle
This is true in general. I don't know a reference for the statement, but it is pretty simple just to work it out. The point is that $L^2(M,E)$ is a Hilbert space which contains $H_0$ as a dense line …
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13
votes
1
answer
1k
views
2-cocycle twists of braided Hopf algebras
2-cocycle twists of Hopf algebras
Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map
$$ f: H \otimes H \to k$$
such that
$$ f(x_{(1)},y_{(1)})f(x_{(2)} y_{( …
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7
votes
1
answer
287
views
Real forms of Drinfeld-Jimbo quantum groups
A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and antipo …
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11
votes
Accepted
what is a spinor structure?
Chapter 9 of Elements of Noncommutative Geometry, by Gracia-Bondia, Varilly, and Figueroa, has this perspective on spin$^c$ and spin structures.
The way to think about this algebraically is that th …
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8
votes
Accepted
Reference for the Hecke relation for the universal R-matrix
For the Drinfeld-Jimbo quantum universal enveloping algebras, see Proposition 24 of Chapter 8 in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. This relation is just in t …
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7
votes
Accepted
Criterion for nilradical of a maximal parabolic subalgebra to be abelian?
Denote by $\mathfrak{l}$ the Levi factor of the parabolic, so that $\mathfrak{p} = \mathfrak{l} \oplus \mathfrak{n}$, and note that this is a splitting as $\mathfrak{l}$-modules. Also denote by $\mat …
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3
votes
Accepted
Lebesgue integral with respect to vector measures?
Marc Rieffel has some notes that develop integration with respect to Banach-space valued measures from the ground up. The notes are very thorough. They are available here:
http://math.berkeley.edu/~ …
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4
votes
Accepted
R-matrices, crystal bases, and the limit as q -> 1
I never found a precise reference for the statement about the R-matrix, so I ended up writing it up myself. The precise statements and proofs can be found in $\S 4.1$ of my paper with Alex Chirvasitu …
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11
votes
1
answer
761
views
R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of q …
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