Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Representation of the Quantum Plane, its Quantum Double and Harmonic
  Analysis on $GL_q^+(2,R)$   We give complete detail of the description of the GNS representation of the
quantum plane $\cA$ and its dual $\hat{\cA}$ as a von-Neumann algebra. In
particular we obtain a rather surprising result that the multiplicative unitary
$W$ is manageable in this quantum semigroup context. We study the quantum
double group construction introduced by Woronowicz, and using Baaj and Vaes'
construction of the multiplicative unitary $\bW_m$, we give the GNS description
of the quantum double $\cD(\cA)$ which is equivalent to $GL_q^+(2,\R)$.
Furthermore we study the fundamental corepresentation $T^{\l,t}$ and its matrix
coefficients, and show that it can be expressed by the $b$-Hypergeometric
function. We also study the regular corepresentation and representation induced
by $\bW_m$, and prove that the space of $L^2$ functions on the quantum double
decomposes into the continuous series representation of $U_q(\gl(2,\R))$ with
the quantum dilogarithm $|S_b(Q+2i\a)|^2$ as the Plancherel measure. Finally we
describe certain representation theoretic meaning of integral transforms
involving the quantum dilogarithm function.

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The categories of this paper are: physics
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn