In this second part, I will try to explain briefly my contribution to Quantum Groups. I studied Lusztig’s results on the so-called restricted specialization of Quantum Group at a root of unity. As I indicated in a previous blog post, the case treated in the literature contains some restrictions on $l$, for example $l$ is supposed to be odd. Basically, I generalized most results to an arbitrary value of $l$ (assuming only $l\geq 7$, in order to factorize irreducible modules).

Let’s start with some notations. Let $\mathfrak{g}$ be a simple Lie algebra and $A={\left(a_{{ij}}\right)}_{{1\leq i,j\leq n}}$ its Cartan matrix. This matrix is symmetrizable and we denote by $D={(d_{i})}_{{1\leq i\leq n}}$ the associated diagonal matrix. Finally $\alpha _{1},\alpha _{2},...,\alpha _{n}$ are the simple roots of $\mathfrak{g}$ and $P$ the weight lattice. As it is well-known, we can consider the universal envelopping algebra $U(\mathfrak{g})$ which has the same representation theory as $\mathfrak{g}$.

The first step is to define the quantum enveloping algebras $U_{q}(\mathfrak{g})$ as a deformation of $U(\mathfrak{g})$. This is an algebra over ${\mathbb{Q}}(q)$, the field of rational functions in the parameter $q$. We let $q_{i}=q^{{d_{i}}}$ and for any integer $n\in{\mathbb{Z}}$, we define a quantum version ${\left[{n}\right]}_{{q_{i}}}=\frac{q_{i}^{n}-q_{i}^{{-n}}}{q_{i}-q_{i}^{{-1}}}$ which obviously converges to the classical $n$ when $q\rightarrow 1$. Similarly, one defines quantum versions ${\left[{n}\right]}_{{q_{i}}}!$, ${\genfrac{[}{]}{0.0pt}{}{n}{k}}_{{q_{i}}}$ of the classical factorial and binomial coefficient. Then we define $U_{q}(\mathfrak{g})$ by generators $X_{i}^{\pm},K_{i}^{{\pm 1}}$ and relations $K_{i}K_{j}=K_{j}K_{i}$, $K_{i}K_{i}^{{-1}}=K_{i}^{{-1}}K_{i}=1$, $K_{i}X_{j}^{\pm}K_{i}^{{-1}}=q_{i}^{{\pm a_{{ij}}}}X_{j}^{\pm}$,

 $[X_{i}^{+},X_{j}^{-}]=X_{i}^{+}X_{j}^{-}-X_{j}^{-}X_{i}^{+}=\delta _{{i,j}}\frac{K_{i}-K_{i}^{{-1}}}{q_{i}-q_{i}^{{-1}}}$

and for $i\neq j$,

 $\sum _{{r=0}}^{{1-a_{{ij}}}}(-1)^{r}{\genfrac{[}{]}{0.0pt}{}{1-a_{{ij}}}{r}}_{{q_{i}}}(X_{i}^{\pm})^{{1-a_{{ij}}-r}}X_{j}^{\pm}(X_{i}^{\pm})^{r}=0$

If we think ”$K_{i}=e^{{(q-1)d_{i}H_{i}}}$”, a first order expansion of the above relations at $q\rightarrow 1$ gives the well-known Chevalley-Serre relations of $U(\mathfrak{g})$. Actually, $U_{q}(\mathfrak{g})$ has the classical properties of $U(\mathfrak{g})$, especially for the representation theory.

Now, we would like to specialize $q$ at an arbitrary complex number $\epsilon$. This makes sense if $\epsilon\neq 0$ and is not a root of unity and in that case we again find a representation theory similar to the classical case. With some technical work, one can still define a specialization at a root of unity $\epsilon$ of order $l$. To do that, one defines for all $k\in{\mathbb{N}}$, the reduced power ${(X_{i}^{\pm})}^{{(k)}}={\left(X_{i}^{\pm}\right)}^{k}/{\left[{k}\right]}_{{q_{i}}}!$ and $\mathcal{A}={\mathbb{Z}}[q,q^{{-1}}]$ the ring of Laurent polynomials. Then $U_{\mathcal{A}}$ is the sub-$\mathcal{A}$-algebra of $U_{q}(\mathfrak{g})$ generated by the $K_{i}^{{\pm 1}}$ and the ${(X_{i}^{\pm})}^{{(k)}}$. One can define a $\mathcal{A}$-basis of $U_{\mathcal{A}}$ (this is not trivial) from which we deduce the restricted specialization ${U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})$.

At that point, one naturally wonders where we have a singularity, i.e. given $k\in{\mathbb{Z}}$, when does $\epsilon _{i}^{k}-\epsilon _{i}^{{-k}}=0$ happen? Using basic arithmetic, that’s the case if and only if $k\equiv 0\mod m_{i}$ where $m=l/\gcd(l,2)$ and $m_{i}=m/\gcd(m,d_{i})$. These $m_{i}$’s act like periods: we get ${\left(X_{i}^{\pm}\right)}^{{m_{i}}}=0$ (when $m$ is greater than one) and ${\left(K_{i}\right)}^{{2m_{i}}}=1$. As a consequence, the important elements to consider become $X_{i}^{\pm}$, ${(X_{i}^{\pm})}^{{(m_{i})}}$, $K_{i}$ (which generate the whole algebra) as well as ${\genfrac{[}{]}{0.0pt}{}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}}=\left[{(X_{i}^{+})}^{{(m_{i})}},{(X_{i}^{-})}^{{(m_{i})}}\right]$ (which, together with the $K_{i}$, form a ”Cartan subalgebra”). Note that if $d_{i}$ is prime to $m$ then $m_{i}=m$ and if moreover $l$ is odd, $m_{i}=l$. Not surprisingly, these are exactly assumptions used in the literature. The only difference in the most general case is that new signs ${(-1)}^{{m_{i}+1}}$, $\epsilon _{i}^{{m_{i}}}={(-1)}^{{(l+1)d_{i}/\delta _{i}}}$ appear in the expressions.

If $V$ is a finite-dimensional ${U_{\epsilon}^{{\mathrm{res}}}}$-module, we can define for all $\sigma\in{\{-1,1\}}^{n}$ and $\lambda\in P$ the weight space

 $\displaystyle V_{{\sigma,\lambda}}=\bigcap _{{1\leq i\leq n}}{\operatorname{Ker}{\mathop{\left(K_{i}-\sigma _{i}\epsilon _{i}^{{\lambda(\alpha _{i}^{\vee})}}1\right)}}\cap\bigcup _{{T\geq 1}}\operatorname{Ker}{\mathop{{\left({\genfrac{[}{]}{0.0pt}{}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}}-\Delta _{i}(\lambda)\left\lfloor\frac{\lambda(\alpha _{i}^{\vee})}{m_{i}}\right\rfloor 1\right)}^{{T}}}}}$

where $\Delta _{i}(\lambda)={(-1)}^{{m_{i}+1}}\sigma _{i}^{{m_{i}}}{(\epsilon _{i}^{{m_{i}}})}^{{\lambda(\alpha _{i}^{\vee})+1}}$. Then $V=\bigoplus _{{\sigma _{,}\lambda}}V_{{\sigma,\lambda}}$ and $X_{j}^{\pm}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm\alpha _{j}}}$ ; ${(X_{j}^{\pm})}^{{(m_{j})}}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm m_{j}\alpha _{j}}}$. In the same way as for the $U_{q}(\mathfrak{g})$-modules, we deduce a classification of irreducible ${U_{\epsilon}^{{\mathrm{res}}}}$-modules. They are all isomorphic to some ${V_{\epsilon}^{{\mathrm{res}}}}(\sigma,\lambda)$ for a type $\sigma$ and a highest weight $\lambda\in P^{+}$. Note that the expression above is not too surprising: if we specialize a $U_{q}(\mathfrak{g})$-module and consider the euclidean division of $\lambda(\alpha _{i}^{\vee})$ by $m_{i}$ then intuitively the remainder goes in the $K_{i}$ part while the quotient goes in the ${\genfrac{[}{]}{0.0pt}{}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}}$ part. But specializing an irreducible $U_{q}(\mathfrak{g})$-module does not necessarily provide an irreducible ${U_{\epsilon}^{{\mathrm{res}}}}$-module and actually one can get by this method a reducible ${U_{\epsilon}^{{\mathrm{res}}}}$-module which is not completely reducible. This latter property is a major difference with the classical case where all representations are semisimple.

Finally, thanks to the Hopf algebra structure of ${U_{\epsilon}^{{\mathrm{res}}}}$, one can define tensor products of ${U_{\epsilon}^{{\mathrm{res}}}}$-modules. We suppose $1\leq d_{i} and consider $\sigma\in\{-1,+1\}^{n},\lambda\in P^{+}$. We note $\lambda=\lambda _{0}+\lambda _{1}$ where for all $1\leq i\leq n$, $0\leq\lambda _{0}(\alpha _{i}^{\vee}) and $\lambda _{1}(\alpha _{i}^{\vee})\equiv 0\mod m_{i}$. Then

 ${V_{\epsilon}^{{\mathrm{res}}}}(\sigma,\lambda)\cong{V_{\epsilon}^{{\mathrm{res}}}}(\sigma,0)\otimes{V_{\epsilon}^{{\mathrm{res}}}}(1,\lambda _{0})\otimes{V_{\epsilon}^{{\mathrm{res}}}}(1,\lambda _{1})$

${V_{\epsilon}^{{\mathrm{res}}}}(\sigma,0)$ is one-dimensional and ${V_{\epsilon}^{{\mathrm{res}}}}(1,\lambda _{0})$ becomes a ${U_{\epsilon}^{{\mathrm{fin}}}}$-module where ${U_{\epsilon}^{{\mathrm{fin}}}}$ is the finite-dimensional subalgebras of ${U_{\epsilon}^{{\mathrm{res}}}}$ generated by the $X_{i}^{\pm},K_{i}^{{\pm 1}}$. What about ${V_{\epsilon}^{{\mathrm{res}}}}(1,\lambda _{1})$? Lusztig’s answer in the case $l$ odd and prime with the $d_{i}$’s is that it is the pullback of the irreducible $U(\mathfrak{g})$-module $V(\lambda _{1}/l)$ by an algebra morphism $\mathrm{Fr}:{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})\rightarrow U(\mathfrak{g})$. This does not seem to be possible in general and I suggested instead to consider the pullback of an irreducible $U_{{\epsilon^{{\prime}}}}(\mathfrak{g})$-module by a morphism $\mathrm{Fr}:{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})\rightarrow U_{{\epsilon^{{\prime}}}}(\mathfrak{g})$ where the order $l^{{\prime}}$ of $\epsilon^{{\prime}}$ is not too big. For example $\epsilon^{{\prime}}=1$ works in Lusztig’s framework, which is consistent with the fact that $U_{1}(\mathfrak{g})$ is close to $U(\mathfrak{g})$. In general, you need to consider the parity of $l$, whether it is a multiple of a $d_{i}$ and also whether the 2-adic order of $m$ is 0, 1 or 2. So I conjectured that $l^{{\prime}}$ can always be taken among the divisors of 24. I explained how to get this result under the assumption that $\mathrm{Fr}$ exists. There is a natural way one would like to define such a morphism but because I did not study in details the $\mathcal{A}$-basis of $U_{\mathcal{A}}$ mentioned above, I can not state this for sure.

Apart from how to build $\mathrm{Fr}$ precisely, my work opens new perspectives. According to a paper of Sawin, the new cases $l$ even and multiple of $d_{i}$ are exactly those used for applications in physics and constructions of topological invariants. Hence my work could give new results in these areas. Also, one can rely on my work to study other generalizations: use of a symetrizable Kac-Moody algebra $\mathfrak{g}$, tilting modules for an arbitrary order $l$, non-restricted specialization for an arbitrary order $l$ etc