# Tagged Questions

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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### Reference request: Principal series are equal in the Grothendieck group

In the usual setup, consider the category of Harish-Chandra $(\mathfrak{g},K)$-modules with given central character (if the central character is regular, this is equivalent to $K$-equivariant $D$-...
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### References of an operator $T: V \otimes V \to V \otimes V$

Let $V$ be a vector space with a basis $v_1, \ldots, v_n$ and let $X_{ij} = v_i \otimes v_j$. Then $X_{ij}, i,j=1,\ldots, n$, is a basis of $V \otimes V$. Let $T: V \otimes V \to V \otimes V$ be the ...
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### Distributions and functions on the Jacquet module $C_c^\infty(X)_{H,\chi}$

Let $X$ be an $\ell$ space (in the sense of Bernstein-Zelevinski), $H$ be an $\ell$ group which acts on $X$ and $\chi$ be a character of $H$. Denote $C^\infty(X)^{H,\chi}$ the space of locally ...
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### Eigenspaces of the Laplace operator on a unit ball

I am interested in structures of the eigenspaces of the Laplace operator on the $n$-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal ...
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### How to obtain a classical r-matrix from a quantum R-matrix?

Let $R$ be a quantum R-matrix. Is there a procedure to dequantize $R$ and obtain a classical r-matrix? Thank you very much.
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### How to write a braiding as a matrix?

Let $V$ be the vector representation of $sl_n$. Then $V \otimes V$ is a $U_q(sl_n)$-module. Suppose that a braiding \begin{align} \Psi: V \otimes V \to V \otimes V \end{align} satisfies the ...
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### Is the cross product $A \rtimes H$ a bialgebra?

Let $H$ be a bialgebra and $A$ a $H$-module algebra. The cross product $A \rtimes H$ is defined as follows. As a vector space $A \rtimes H = A \otimes H$. The multiplication on $A \rtimes H$ is ...
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### Is there a matrix representation of the permutation group whose character is the Markov trace?

Let $g$ be an element in the permutation group (symmetric group) $S_N$. Define the Markov trace of $g$ (denoted $\text{tr}_k g$) as $$\text{tr}_kg = k^\text{number of cycles in g} ,$$ which depends ...