I am a fan of the Matrix exponential $\exp(X)$, defined for any complex matrix $X$ by \begin{equation*} \exp(X) := \sum_{k \ge 0} \frac{X^k}{k!}. \end{equation*}

I have a fleeting acquaintance with q-analog's (essentially, I know that they exist, but have almost no idea what use they serve, which is part of the reason why I am asking this question).

Thus, my question is

Has the following q-analog of the matrix exponential \begin{equation*} \exp_q(X) := \sum_{k \ge 0} \frac{X^k}{[k]_q!}, \end{equation*} been studied previously? If so, in what context?

PS: More generally, the above question can be rephrased in terms of q-analogs of functions of matrices (which includes scalar, vector, and matrix valued functions). But I wanted to limit my focus to a concrete case of special interest to me.

  • $\begingroup$ also, please feel free to edit the tags to this post, if needed. thanks. $\endgroup$ – Suvrit Mar 1 '12 at 17:16
  • $\begingroup$ I believe the keyword you want is "Eulerian generating function." $\endgroup$ – Qiaochu Yuan Mar 1 '12 at 17:19
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    $\begingroup$ As you could have guess it is called the "q-exponential" and is quite common e.g. in the theory of quantum groups. Notably, the universal R-matrix of $U_q(\mathfrak g)$ can be written as a product of q-exponential. $\endgroup$ – Adrien Mar 1 '12 at 17:22
  • $\begingroup$ Thanks Adrien, but is $q$ itself a matrix in the quantum case? In that case, probably the above "definition" that I have is not the right one? $\endgroup$ – Suvrit Mar 1 '12 at 17:28
  • $\begingroup$ @Qiaochu: Notice that $X$ is a matrix --- or is your comment valid even for this case? $\endgroup$ – Suvrit Mar 1 '12 at 17:34

It's more an example than a general answer. Details can be found here : http://arxiv.org/abs/math/0512500

It is convenient to replace $q$ by $q^2$ in the formula that you gave. Doing so we have the following desirable identities:

  • $\exp_q(x)\exp_{-q}(-x)=1$
  • if $xy=q^2yx$, then $\exp_q(y)\exp_q(x)=\exp_q(x+y)$

Now if $\mathfrak g$ is a simple Lie algebra, $\Phi^+$ a choice of positive roots, $h_i,e_{\alpha},f_{\alpha}$ the generators of $U_q(\mathfrak g)$ associated to the Chevalley basis of $\mathfrak g$, $>$ a normal ordering on $\Phi^+$, and $q_{\alpha}=q^{(\alpha,\alpha)/2}$, then the R-matrix of $U_q(\mathfrak g)$ is given by

$$ R=K \prod_{\alpha \in \Phi^+}^> R_{\alpha} $$ where $$K=q^{\sum h_i \otimes h^i}$$ and $$R_{\alpha}=\exp_{q_{\alpha}^{-1}}((q_{\alpha}-q_{\alpha}^{-1})e_{\alpha} \otimes f_{\alpha})$$

Here $q$ is either a generic complex number, or a variable, in which case we work over the field $\mathbb{Q}(q^{\frac12})$.

It is a universal formula, but of course you can specialize it to an element of $End(V\otimes V)$ for any $U_q(\mathfrak g)$-module $V$.


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