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 ll, for example ll is supposed to be odd. Basically, I generalized most results to an arbitrary value of ll (assuming only l7l\geq 7, in order to factorize irreducible modules).

Let’s start with some notations. Let 𝔤\mathfrak{g} be a simple Lie algebra and A=(aij)1i,jnA={\left(a_{{ij}}\right)}_{{1\leq i,j\leq n}} its Cartan matrix. This matrix is symmetrizable and we denote by D=(di)1inD={(d_{i})}_{{1\leq i\leq n}} the associated diagonal matrix. Finally α1,α2,,αn\alpha _{1},\alpha _{2},...,\alpha _{n} are the simple roots of 𝔤\mathfrak{g} and PP the weight lattice. As it is well-known, we can consider the universal envelopping algebra U(𝔤)U(\mathfrak{g}) which has the same representation theory as 𝔤\mathfrak{g}.

The first step is to define the quantum enveloping algebras Uq(𝔤)U_{q}(\mathfrak{g}) as a deformation of U(𝔤)U(\mathfrak{g}). This is an algebra over (q){\mathbb{Q}}(q), the field of rational functions in the parameter qq. We let qi=qdiq_{i}=q^{{d_{i}}} and for any integer nn\in{\mathbb{Z}}, we define a quantum version [n]qi=qin-qi-nqi-qi-1{\left[{n}\right]}_{{q_{i}}}=\frac{q_{i}^{n}-q_{i}^{{-n}}}{q_{i}-q_{i}^{{-1}}} which obviously converges to the classical nn when q1q\rightarrow 1. Similarly, one defines quantum versions [n]qi!{\left[{n}\right]}_{{q_{i}}}!, [nk]qi{\genfrac{[}{]}{0.0pt}{}{n}{k}}_{{q_{i}}} of the classical factorial and binomial coefficient. Then we define Uq(𝔤)U_{q}(\mathfrak{g}) by generators Xi±,Ki±1X_{i}^{\pm},K_{i}^{{\pm 1}} and relations KiKj=KjKiK_{i}K_{j}=K_{j}K_{i}, KiKi-1=Ki-1Ki=1K_{i}K_{i}^{{-1}}=K_{i}^{{-1}}K_{i}=1, KiXj±Ki-1=qi±aijXj±K_{i}X_{j}^{\pm}K_{i}^{{-1}}=q_{i}^{{\pm a_{{ij}}}}X_{j}^{\pm},

[Xi+,Xj-]=Xi+Xj--Xj-Xi+=δi,jKi-Ki-1qi-qi-1[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 iji\neq j,

r=01-aij(-1)r[1-aijr]qi(Xi±)1-aij-rXj±(Xi±)r=0\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 ”Ki=e(q-1)diHiK_{i}=e^{{(q-1)d_{i}H_{i}}}”, a first order expansion of the above relations at q1q\rightarrow 1 gives the well-known Chevalley-Serre relations of U(𝔤)U(\mathfrak{g}). Actually, Uq(𝔤)U_{q}(\mathfrak{g}) has the classical properties of U(𝔤)U(\mathfrak{g}), especially for the representation theory.

Now, we would like to specialize qq at an arbitrary complex number ϵ\epsilon. This makes sense if ϵ0\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 ϵ\epsilon of order ll. To do that, one defines for all kk\in{\mathbb{N}}, the reduced power (Xi±)(k)=(Xi±)k/[k]qi!{(X_{i}^{\pm})}^{{(k)}}={\left(X_{i}^{\pm}\right)}^{k}/{\left[{k}\right]}_{{q_{i}}}! and 𝒜=[q,q-1]\mathcal{A}={\mathbb{Z}}[q,q^{{-1}}] the ring of Laurent polynomials. Then U𝒜U_{\mathcal{A}} is the sub-𝒜\mathcal{A}-algebra of Uq(𝔤)U_{q}(\mathfrak{g}) generated by the Ki±1K_{i}^{{\pm 1}} and the (Xi±)(k){(X_{i}^{\pm})}^{{(k)}}. One can define a 𝒜\mathcal{A}-basis of U𝒜U_{\mathcal{A}} (this is not trivial) from which we deduce the restricted specialization Uϵres(𝔤){U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}).

At that point, one naturally wonders where we have a singularity, i.e. given kk\in{\mathbb{Z}}, when does ϵik-ϵi-k=0\epsilon _{i}^{k}-\epsilon _{i}^{{-k}}=0 happen? Using basic arithmetic, that’s the case if and only if k0modmik\equiv 0\mod m_{i} where m=l/gcd(l,2)m=l/\gcd(l,2) and mi=m/gcd(m,di)m_{i}=m/\gcd(m,d_{i}). These mim_{i}’s act like periods: we get (Xi±)mi=0{\left(X_{i}^{\pm}\right)}^{{m_{i}}}=0 (when m is greater than one) and (Ki)2mi=1{\left(K_{i}\right)}^{{2m_{i}}}=1. As a consequence, the important elements to consider become Xi±X_{i}^{\pm}, (Xi±)(mi){(X_{i}^{\pm})}^{{(m_{i})}}, KiK_{i} (which generate the whole algebra) as well as [Ki;0mi]ϵi=[(Xi+)(mi),(Xi-)(mi)]{\genfrac{[}{]}{0.0pt}{}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}}=\left[{(X_{i}^{+})}^{{(m_{i})}},{(X_{i}^{-})}^{{(m_{i})}}\right] (which, together with the KiK_{i}, form a ”Cartan subalgebra”). Note that if did_{i} is prime to mm then mi=mm_{i}=m and if moreover ll is odd, mi=lm_{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)mi+1{(-1)}^{{m_{i}+1}}, ϵimi=(-1)(l+1)di/δi\epsilon _{i}^{{m_{i}}}={(-1)}^{{(l+1)d_{i}/\delta _{i}}} appear in the expressions.

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

Vσ,λ=1inKer(Ki-σiϵiλ(αi)1)T1Ker([Ki;0mi]ϵi-Δi(λ)λ(αi)mi1)T\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 Δi(λ)=(-1)mi+1σimi(ϵimi)λ(αi)+1\Delta _{i}(\lambda)={(-1)}^{{m_{i}+1}}\sigma _{i}^{{m_{i}}}{(\epsilon _{i}^{{m_{i}}})}^{{\lambda(\alpha _{i}^{\vee})+1}}. Then V=σ,λVσ,λV=\bigoplus _{{\sigma _{,}\lambda}}V_{{\sigma,\lambda}} and Xj±VσλVσ,λ±αjX_{j}^{\pm}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm\alpha _{j}}} ; (Xj±)(mj)VσλVσ,λ±mjαj{(X_{j}^{\pm})}^{{(m_{j})}}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm m_{j}\alpha _{j}}}. In the same way as for the Uq(𝔤)U_{q}(\mathfrak{g})-modules, we deduce a classification of irreducible Uϵres{U_{\epsilon}^{{\mathrm{res}}}}-modules. They are all isomorphic to some Vϵres(σ,λ){V_{\epsilon}^{{\mathrm{res}}}}(\sigma,\lambda) for a type σ\sigma and a highest weight λP+\lambda\in P^{+}. Note that the expression above is not too surprising: if we specialize a Uq(𝔤)U_{q}(\mathfrak{g})-module and consider the euclidean division of λ(αi)\lambda(\alpha _{i}^{\vee}) by mim_{i} then intuitively the remainder goes in the KiK_{i} part while the quotient goes in the [Ki;0mi]ϵi{\genfrac{[}{]}{0.0pt}{}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}} part. But specializing an irreducible Uq(𝔤)U_{q}(\mathfrak{g})-module does not necessarily provide an irreducible Uϵres{U_{\epsilon}^{{\mathrm{res}}}}-module and actually one can get by this method a reducible Uϵres{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 Uϵres{U_{\epsilon}^{{\mathrm{res}}}}, one can define tensor products of Uϵres{U_{\epsilon}^{{\mathrm{res}}}}-modules. We suppose 1di<m1\leq d_{i}<m and consider σ{-1,+1}n,λP+\sigma\in\{-1,+1\}^{n},\lambda\in P^{+}. We note λ=λ0+λ1\lambda=\lambda _{0}+\lambda _{1} where for all 1in1\leq i\leq n, 0λ0(αi)<mi0\leq\lambda _{0}(\alpha _{i}^{\vee})<m_{i} and λ1(αi)0modmi\lambda _{1}(\alpha _{i}^{\vee})\equiv 0\mod m_{i}. Then

Vϵres(σ,λ)Vϵres(σ,0)Vϵres(1,λ0)Vϵres(1,λ1){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})

Vϵres(σ,0){V_{\epsilon}^{{\mathrm{res}}}}(\sigma,0) is one-dimensional and Vϵres(1,λ0){V_{\epsilon}^{{\mathrm{res}}}}(1,\lambda _{0}) becomes a Uϵfin{U_{\epsilon}^{{\mathrm{fin}}}}-module where Uϵfin{U_{\epsilon}^{{\mathrm{fin}}}} is the finite-dimensional subalgebras of Uϵres{U_{\epsilon}^{{\mathrm{res}}}} generated by the Xi±,Ki±1X_{i}^{\pm},K_{i}^{{\pm 1}}. What about Vϵres(1,λ1){V_{\epsilon}^{{\mathrm{res}}}}(1,\lambda _{1})? Lusztig’s answer in the case ll odd and prime with the did_{i}’s is that it is the pullback of the irreducible U(𝔤)U(\mathfrak{g})-module V(λ1/l)V(\lambda _{1}/l) by an algebra morphism Fr:Uϵres(𝔤)U(𝔤)\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 Uϵ(𝔤)U_{{\epsilon^{{\prime}}}}(\mathfrak{g})-module by a morphism Fr:Uϵres(𝔤)Uϵ(𝔤)\mathrm{Fr}:{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})\rightarrow U_{{\epsilon^{{\prime}}}}(\mathfrak{g}) where the order ll^{{\prime}} of ϵ\epsilon^{{\prime}} is not too big. For example ϵ=1\epsilon^{{\prime}}=1 works in Lusztig’s framework, which is consistent with the fact that U1(𝔤)U_{1}(\mathfrak{g}) is close to U(𝔤)U(\mathfrak{g}). In general, you need to consider the parity of ll, whether it is a multiple of a did_{i} and also whether the 2-adic order of mm is 0, 1 or 2. So I conjectured that ll^{{\prime}} can always be taken among the divisors of 24. I explained how to get this result under the assumption that Fr\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 𝒜\mathcal{A}-basis of U𝒜U_{\mathcal{A}} mentioned above, I can not state this for sure.

Apart from how to build Fr\mathrm{Fr} precisely, my work opens new perspectives. According to a paper of Sawin, the new cases ll even and multiple of did_{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 𝔤\mathfrak{g}, tilting modules for an arbitrary order ll, non-restricted specialization for an arbitrary order ll etc