Quantum groups and Lyndon words

## Quantum groups and Lyndon words

In these three talks, I will explore the connections between

1. Mirkovic-Vilonen polytopes,
2. the Littelmann path model,
3. quantum groups,
4. free Lie algebras,
5. words.

The lectures will proceed as follows:

1. Combinatorics
2. Algebra
3. Geometry

Let $ℐ$ be an alphabet, and let

Let $𝒫$ be the free associative algebra generated by a set of elements ${p}_{i},$ indexed by the elements $i\in ℐ.$

$ℒ$ is generated by and with a bracket operation $\left[\right]:ℒ×ℒ\to ℒ.$ There are many possible bracket operations:

1. $\left[u,v\right]=uv-vu.$
2. $\left[u,v\right]=uv-{q}^{⟨\mathrm{deg}\left(u\right),\mathrm{deg}\left(v\right)⟩}vu.$

Example: Consider the alphabet $ℐ=\left\{+,-\right\}.$ A typical word will be ++-+--$\in ℐ.$ A typical element of $𝒫$ will be ${p}_{+}{p}_{-}-2{p}_{-}{p}_{+}\in 𝒫.$ If the bracket is $\left[u,v\right]=uv-vu,$ then $[ p+ , p- ]= p+ p- - p- p+ ,$ and $\left[{p}_{+},{p}_{-}\right]=-\left[{p}_{-},{p}_{+}\right]$

$𝒫$ is the enveloping algebra of $ℒ.$ Choose a total order on $ℐ$ and then use the lexicographic order on ${ℐ}^{*}.$ is the set of Lyndon words. For $l\in L$ define

Example: (cont) There are two possible choices for the ordering: $+<-$ or $-<+.$ If $+<-$ then $+-\in L$ but $-+\ne L.$ Then $\left[+\right]={p}_{+},\left[-\right]={p}_{-}$ and $\left[+-\right]=\left[\left[-\right],\left[+\right]\right]=\left[{p}_{-},{p}_{+}\right]$

1. If $u\in {I}^{*}$ then $u$ has a unique factorisation $u={l}_{1}{l}_{2}\dots {l}_{n}$ with ${l}_{1},\dots ,{l}_{n}\in L,{l}_{1}\ge \dots \ge {l}_{n}.$
2. $ℒ$ has basis $\left\{\left[l\right]\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}l\in L\right\}$ and $𝒫$ has basis $\left\{\left[{l}_{1}\right]\dots \left[{l}_{n}\right]\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{l}_{1},\dots ,{l}_{n}\in L,{l}_{1}\ge \dots \ge {l}_{n}\right\}.$

## The quantum group ${U}^{-}$

$𝒫$ is graded by
Example:

1. (Type ${A}_{2}$) $ℐ=\left\{+,-\right\}.$ Then $αi αj = 2 -1 -1 2$
2. (Type ${B}_{n}$)$ℐ=\left\{1,2,\dots ,n\right\},$ then $αi αj = 2 -2 0 -2 4 -2 -2 4 … … … -2 0 -2 4$

Fix a symmetric bilinear form $⟨,⟩:{Q}^{+}×{Q}^{+}\to ℤ$ given by values $,⟨{\alpha }_{i},{\alpha }_{j}⟩\in ℤ.$

The dual of $𝒫$ is with The $⟨,⟩$-shuffle product on $ℱ$ is the sum is over shuffles of $u$ with $v,$ $wt σuv = ∑ 1≤iσ j - u i v k-j$ where ${u}_{i}$ is the $i$th letter in $u$ and ${v}_{k-j}$ is the $\left(k-j\right)$th letter in $v.$

${U}^{-}$ i the $·$-subalgebra of $ℱ$ generated by ${f}_{i},i\in I.$

## Good Lyndon words

$U - ↪ ℱ f i ↦ f i and 𝒫 ↠ U - p i ↦ f i$ and the restriction of $⟨,⟩:𝒫×ℱ\to ℂ$ to ${U}^{-},$$, : U - × U - →ℂ$ is nondegenerate.

Example (cont) HW: Show that ${f}_{+}\circ {f}_{+}\circ {f}_{+}-\left(q+{q}^{-1}\right){f}_{+}\circ {f}_{-}\circ {f}_{+}+{f}_{-}\circ {f}_{+}\circ {f}_{+}=0$ in ${U}^{-}.$

The good words and the good Lyndon words are $\mathrm{GL}=G\cap L$ and

HW: $\begin{array}{rcl}{f}_{+}\circ {f}_{+}\circ {f}_{-}& =& {f}_{+}\circ \left({f}_{+}{f}_{-}+q{f}_{-}{f}_{+}\right)\\ & =& {f}_{+}{f}_{+}{f}_{-}+{q}^{-2}{f}_{+}{f}_{+}{f}_{-}\dots \end{array}$ The shuffles ${f}_{+}\circ {f}_{+}\circ {f}_{-}$ and ${f}_{+}\circ {f}_{-}\circ {f}_{+}$ and ${f}_{-}\circ {f}_{+}\circ {f}_{+}$ are not independent and we must choose preferred words for a basis of ${U}^{-}.$$f + ∘ f - = f + f - +q f - f +$ so ${f}_{-}{f}_{+}\in G.$

Problem: Describe $G$ for $2 -2 -2 2 .$

(Lalonde-Ram, Leclerc)

1. Assume that $α i ∨ α j with α i ∨ = 2 αi αi αi$ is a symmetrisable Cartan matrix. Let ${ℛ}^{+}$ be the positive roots of the corresponding Kac-Mody Lie algebra $𝔤.$ $GL → ℛ + l ↦ deg l$ is a bijection with inverse $ℛ + → GL β ↦ l β$ given by $l β =max l β 1 l β2 | β1 , β2 ∈ ℛ + , β1 + β2 ∈β,l β1
2. PBW basis and canonical basis of ${U}^{-}$
${U}^{-}$ has basis $\left\{{E}_{g}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in G\right\},$ where
3. The ordering on ${ℛ}^{+}$ induced by lexicographic ordering on $\mathrm{GL}$ determines a reduced ecomposition of ${w}_{0},$ $w 0 = s i 1 … s i N ,so that ℛ + = β1 <… βN$ with $β1 = α i 1 , β2 = s i 1 α i 2 ,…, βN = s i N α i N .$ Let $-:{U}^{-}\to {U}^{-}$ be the automorphism given by $f i - = f i and q - = q -1 .$ The canonical basis of ${U}^{-}$ is $\left\{{b}_{g}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in G\right\}$ given by $b g - = b g and b g = E g + ∑ h∈G,h>g c gh E h ,$ with ${c}_{gh}\in qℤ\left[q\right].$

HW: If $+<-$ then $G\cap L=\left\{+,-,+-\right\}$ and ${ℛ}^{+}=\left\{{\alpha }_{+},{\alpha }_{-},-{\alpha }_{+}+{\alpha }_{-}\right\}$

## Dual PBW and dual canonical bases

The dual PBW basis of ${U}^{-}$ is $\left\{{E}_{g}^{*}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in G\right\}$ given by $⟨{E}_{g},{E}_{h}^{*}⟩={\delta }_{gh}.$

1. (Lusztig) ${E}_{h}^{*}=\left(\text{const}\right){E}_{h}.$
2. (Leclerc) $\left\{{b}_{g}^{*}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}g\in G\right\}$ is characterised by and ${U}^{-}$ has bases $PBW basis:: E g | g∈ G → ,$ $dual PBW basis:: E g * | g∈ G → ,$$canonical basis:: b g | g∈ G → ,$ $dual canonical basis:: b g * | g∈ G → .$