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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 quantum groups. I am very new to crystal bases, so I would also appreciate corrections if my questions are not well-formulated. I am putting the questions first, followed by the motivation for those who are curious.


Do you know references (with proofs) for the following statements:

  1. Let $V$ be a finite-dimensional $U_q(\mathfrak{g})$-module. Then the divided powers $E_{\beta}^{(t)}, F_\beta^{(t)}$ of the quantum root vectors have matrix coefficients given by Laurent polynomials in $q$, with respect to the global crystal basis for $V$.

  2. Let $V,W$ be finite-dimensional $U_q(\mathfrak{g})$-modules. Then the matrix coefficients of the $R$-matrix $R_{V,W}$ are Laurent polynomials in $q$, with respect to the tensor product of the global crystal bases for $V,W$.

I believe I have proofs for these statements, but it would be nice to just reference something definitive instead of writing the proofs out myself.


Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of $\mathfrak{g}$ for $q$ not a root of unity, with generators $E_i,F_i,K_i$ corresponding to the simple roots of $\mathfrak{g}$. Using an action of the braid group of $\mathfrak{g}$ on $U_q(\mathfrak{g})$ one can define quantum root vectors $E_\beta,F_\beta$ for all positive roots $\beta$. (This depends on a choice of decomposition of the longest word of the Weyl group, so assume that we have fixed such a decomposition.)

Let $R_{U,V}$ be the action of the $R$-matrix on $U \otimes V$, (as in Chari-Pressley or Klimyk-Schmudgen, say) so $\tau \circ R_{U,V}$ is the braiding. I would like to make sense of the statement that $R_{V,W} \to \mathrm{id}_{V \otimes W}$ as $q \to 1$ (and hence the braiding tends to the flip as $q \to 1$). This is not trivial because for different $q$'s, the operators $R_{V,W}$ are really operators on different vector spaces. This is where the crystal bases come in.

As I understand it, a crystal basis for a module has the property that the matrix coefficients of the generators $E_i,F_i$ of $U_q(\mathfrak{g})$ (and divided powers of the generators) are given by universal Laurent polynomials in $q$ whose coefficients are independent of $q$. Using this basis we can think of all of the algebras $U_q(\mathfrak{g})$ for various $q$'s acting on the same vector space. The point is that Laurent polynomials are continuous and well-defined at $q=1$, i.e. they are specializable to $q=1$.

Taking the tensor product of the crystal bases for $V$ and $W$, we can think of all of the $R$-matrices for various $q$'s acting on the same space as well, and it makes sense to ask if this family of $R$-matrices is continuous in $q$, and if so, whether it can be extended to $q=1$.

The formula for the action of the $R$-matrix is a big sum of products of operators of the form

$$ \frac{1}{[t]_{q_\beta}!} E_\beta^t \otimes F_\beta^t$$

with coefficients given by Laurent polynomials in $q$. Putting the $q$-factorial under, say, the $E_\beta^t$ term gives the divided power $E_\beta^{(t)}$. If the quantum root vectors and their divided powers act by Laurent polynomials, then the $R$-matrix does as well, and hence everything in sight is continuous in $q$, can be specialized to $q=1$, and it is clear that at $q=1$ the $R$-matrix is just the identity.

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up vote 4 down vote accepted

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, Remarks on quantum symmetric algebras, available here.

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