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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.

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 …

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 …

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 …

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 …

I realy like the exercises in Gert Pedersen's book Analysis Now.

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 …

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 …

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 …

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 …

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 …

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_{( …

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 …

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/~ …

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 …