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Thursday, September 13 2012

Master Thesis, LaTeXML and Quantum Groups (part 2)

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

Master Thesis, LaTeXML and Quantum Groups (part 1)

I wanted to wait for my oral defense before blogging about my master thesis and how I manage to publish Web and paper versions of it. However, I finally met my supervisor today and the oral defense is only likely to take place next Wednesday. I do not want to delay too much this blog post and thus decided to publish it today...

This year, I have taken the following approach to write my master thesis: from LaTeX input files, I used XeLaTeX to generate a pdf document and LaTeXML to generate HTML+MathML Web pages. I had to handle some small differences via separate configuration files. However in general, these two tools are compatible and accept more or less the same LaTeX input. I did not really have to make graphics: I only used the amscd package to draw simple commutative diagrams and did not try to draw schemas for representations of quantum groups. Hence I did not get the opportunity to test how LaTeXML can generate MathML inside SVG, although I saw on July something interesting for Firefox on the LaTeXML mailing list.

The pdf version provides a good print layout which allows to workaround some issues that I had two years ago when I printed my Master Thesis in Computer Science directly from Firefox. XeLaTeX also seems much faster and so more convenient to use when you only want to check that your LaTeX code is syntactically correct and get a quick preview. It seems that XeLaTeX uses a kind of cache: there are intermediary files that I guess are used again when you regenerate the document. In contrast, LaTeXML seems to always regenerate one big XML file in a first step and the Web pages in a second step. Perhaps LaTeXML has an option to avoid that behavior or perhaps the idea of a cache system does not work well in the case of Web pages.

The output of LaTeXML has the classical advantages of HTML+MathML for publication on the Web and is much more comfortable to read on a screen. Generally speaking, I think Firefox renders pretty well the LaTeXML output. LaTeXML generates HTML rows to implement labelling and does not rely on mathvariant, which allow to avoid issues with <mlabeledtr> and token elements. However, I still note some MathML's rendering imperfections which, not surprisingly, have already be mentioned in the MathML project roadmap:

  • Linebreaking: bad line breaks inside some equations, apparently those generated by some environments like multline or gathered. Sometimes, I also see bad line breaks around equations for example when they are inside parenthesis.
  • Spacing: the lack of support for mtable@rowspacing/columnspacing seems to give wrong spacing inside binom-like notations. For some reason LaTeXML generates <mpadded> elements of zero width in some places and they cause weird overlappings in some summations.
  • Operator Stretching: commutative diagrams in the definition of Hopf algebras would look better if we support stretching operators in table cells.

This also gives me the opportunity to report various bugs and give some suggestions to the LaTeXML team, including the use of MathJax (for browsers without MathML support), the replacement of <mfenced> by the equivalent <mrow>, <mo> constructions (better rendering in Firefox), improvement to the generation of headers in HTML5 and more.

The title of my master thesis is "Specialization of Quantum Groups at a Root of Unity and Finite Dimensional Representations". The concept of Quantum Groups is based on ideas from theoretical physics, but I studied these structures from a purely algebraic point of view. That is not likely to be interesting if you have never heard about Lie algebras or are not familiar with representation theory, so I will present my contribution in a separate blog post.

Saturday, June 23 2012

Quantum Groups at root of unity

All the papers or books I read so far on quantum groups about Lusztig’s restricted specialization consider only primitive roots of unity of odd order and additional conditions. In most cases, it is claimed that these restrictions could be removed without too much harm but details are not given. So I have tried to do the calculation myself. Below is what I find for the finite dimensional representations of Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}). Note that most authors consider the case l=m=mil=m=m_{i} odd. Then -1mi=ϵimi=1{(-1)}^{{m_{i}}}=\epsilon _{i}^{{m_{i}}}=1 and we can even restrict the study to the representations for which σ\sigma is always 1. Indeed, all these assumptions make the expression below much simpler.

We consider g\mathfrak{g} a simple Lie algebra of rank nn and use standard notations αi\alpha _{i} for roots, αi\alpha _{i}^{\vee} for coroots and PgP(\mathfrak{g}) for the weight space. Let l>2l>2 be an integer and ϵ\epsilon a primitive ll-root of the unity. We let m=l2m=\frac{l}{2} if ll is even and m=lm=l otherwise. For all 1in1\leq i\leq n, define ϵi=ϵdi\epsilon _{i}=\epsilon^{{d_{i}}}, δi=gcddi,m\delta _{i}=\gcd(d_{i},m), and mi=mδim_{i}=\frac{m}{\delta _{i}}. We assume that no did_{i} is a multiple of mm, which is obviously true for ll large enough.

We denote Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}) the restricted specialization as defined in Chari and Pressley’s guide to quantum groups and keep their notations for the elements KiK_{i}, Ki;crϵi{\genfrac{[}{]}{0.0pt}{0}{{K_{i}};{c}}{r}}_{{\epsilon _{i}}}, Xi±X_{i}^{{\pm}} and Xi±r{(X_{i}^{\pm})}^{{(r)}} of Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}). Finally, we let VV be a finite Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})-module.

For any λPg\lambda\in P(\mathfrak{g}) and σ-1,1n\sigma\in{\{-1,1\}}^{n} define

Vσ,λ=1inKerKi-σiϵiλαi1KerKi;0miϵi-Δiλλαimi1V_{{\sigma,\lambda}}=\bigcap _{{1\leq i\leq n}}{\operatorname{Ker}{\left(K_{i}-\sigma _{i}\epsilon _{i}^{{\lambda(\alpha _{i}^{\vee})}}1\right)}\cap\operatorname{Ker}{{\left({\genfrac{[}{]}{0.0pt}{0}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}}-\Delta _{i}(\lambda)\left\lfloor\frac{\lambda(\alpha _{i}^{\vee})}{m_{i}}\right\rfloor 1\right)}}}

where i,Δiλ=-1mi+1σimiϵimiλαi+1\forall i,\Delta _{i}(\lambda)={(-1)}^{{m_{i}+1}}\sigma _{i}^{{m_{i}}}{(\epsilon _{i}^{{m_{i}}})}^{{\lambda(\alpha _{i}^{\vee})+1}}

For any 1jn1\leq j\leq n, we have

Xj±VσλVσ,λ±αjX_{j}^{\pm}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm\alpha _{j}}}
Xj±mjVσλVσ,λ±mjαj{(X_{j}^{\pm})}^{{(m_{j})}}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm m_{j}\alpha _{j}}}

Moreover if VV is simple, V=σ,λVσ,λV=\bigoplus _{{\sigma,\lambda}}V_{{\sigma,\lambda}}.

\square