Quantum groups and Lyndon words

Data for quiver Hecke algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

Quantum groups and Lyndon words

In these three talks, I will explore the connections between

Mirkovic-Vilonen polytopes,
the Littelmann path model,
quantum groups,
free Lie algebras,
words.

The lectures will proceed as follows:

Combinatorics
Algebra
Geometry

Let $ℐ$ be an alphabet, and let $ℐ^{*} = {words on ℐ} .$

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 $p_{i}, i \in ℐ,$ and with a bracket operation $[] : ℒ \times ℒ \to ℒ .$ There are many possible bracket operations:

$[u, v] = u v - v u .$
$[u, v] = u v - q^{⟨\deg (u), \deg (v)⟩} v u .$

Example: Consider the alphabet $ℐ = {+, -} .$ 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 $[u, v] = u v - v u,$ then $[p_{+}, p_{-}] = p_{+} p_{-} - p_{-} p_{+},$ and $[p_{+}, p_{-}] = - [p_{-}, p_{+}]$

$𝒫$ is the enveloping algebra of $ℒ .$ Choose a total order on $ℐ$ and then use the lexicographic order on $ℐ^{*} .$ $L = \{l \in ℐ^{*} | l \leq l_{2} if l = l_{1} l_{2}\}$ is the set of Lyndon words. For $l \in L$ define $[l] = [[l_{1}], [l_{2}]], if l = l_{1} l_{2} with l_{1} \in L of maximum length \neq l, and [k] = l if l \in I .$

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

If $u \in I^{*}$ then $u$ has a unique factorisation $u = l_{1} l_{2} \dots l_{n}$ with $l_{1}, \dots, l_{n} \in L, l_{1} \geq \dots \geq l_{n} .$
$ℒ$ has basis $\{[l] | l \in L\}$ and $𝒫$ has basis $\{[l_{1}] \dots [l_{n}] | l_{1}, \dots, l_{n} \in L, l_{1} \geq \dots \geq l_{n}\} .$

The quantum group $U^{-}$

$𝒫$ is graded by $Q^{+} = \sum_{i \in I} ℤ_{\geq 0} α_{i} with \deg (p_{i_{1}}, \dots, p_{i_{l}}) = α_{i_{1}} + \dots + α_{i_{l}} .$
Example:

(Type $A_{2}$ ) $ℐ = \{+ -\} .$ Then $(⟨α_{i}, α_{j}⟩) = (\begin{array}{rcl} 2 & - 1 \\ - 1 & 2 \end{array})$
(Type $B_{n}$ ) $ℐ = \{12 \dots n\},$ then $(⟨α_{i}, α_{j}⟩) = (\begin{array}{rcl} \begin{array}{rcl} 2 & - 2 & 0 \\ - 2 & 4 & - 2 \\ - 2 & 4 & \dots \\ \dots & \dots & - 2 \\ 0 & - 2 & 4 \end{array} \end{array})$

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

The dual of $𝒫$ is $ℱ = free algebra generated by f_{i}, i \in I$ with $⟨p_{i}, f_{j}⟩ = δ_{i j}, if p_{i} = p_{i_{1}} \dots p_{i_{k}} and f_{j} = f_{j_{1}} \dots f_{j_{l}} .$ The $⟨,⟩$ -shuffle product on $ℱ$ is $u \cdot v = \sum_{σ \in S_{k + l} / S_{k} \times S_{l}} q^{wt (σ, u, v)} σ (u v), where k = l (u), l = l (v),$ the sum is over shuffles of $u$ with $v,$ $wt (σ, u, v) = \sum_{1 \leq i < j \leq k + l, σ (i) > σ (j)} - ⟨u_{i}, v_{k - j}⟩$ where $u_{i}$ is the $i$ th letter in $u$ and $v_{k - j}$ is the $(k - j)$ th letter in $v .$

$U^{-}$ i the $\cdot$ -subalgebra of $ℱ$ generated by $f_{i}, i \in I .$

Good Lyndon words

$\begin{array}{rcl} U^{-} & ↪ & ℱ \\ f_{i} & \mapsto & f_{i} \end{array} and \begin{array}{rcl} 𝒫 & ↠ & U^{-} \\ p_{i} & \mapsto & f_{i} \end{array}$ and the restriction of $⟨,⟩ : 𝒫 \times ℱ \to ℂ$ to $U^{-},$ $⟨,⟩ : U^{-} \times U^{-} \to ℂ$ is nondegenerate.

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

The good words and the good Lyndon words are $GL = G \cap L$ and $G = \{g \in I^{*} | g is the maximal word in an element of U^{-}\}$

HW: $\begin{array}{rcl} f_{+} \circ f_{+} \circ f_{-} & = & f_{+} \circ (f_{+} f_{-} + q f_{-} f_{+}) \\ = & f_{+} f_{+} f_{-} + q^{- 2} f_{+} f_{+} f_{-} \dots \end{array}$ 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 $U^{-} .$ $f_{+} \circ f_{-} = f_{+} f_{-} + q f_{-} f_{+}$ so $f_{-} f_{+} \in G .$

Problem: Describe $G$ for $(\begin{array}{rcl} 2 & - 2 \\ - 2 & 2 \end{array}) .$

(Lalonde-Ram, Leclerc)

Assume that $(⟨α_{i}^{\lor}, α_{j}⟩) with α_{i}^{\lor} = \frac{2 α_{i}}{⟨α_{i}, α_{i}⟩}$ is a symmetrisable Cartan matrix. Let $ℛ^{+}$ be the positive roots of the corresponding Kac-Mody Lie algebra $𝔤 .$ $\begin{array}{rcl} GL & \to & ℛ^{+} \\ l & \mapsto & \deg (l) \end{array}$ is a bijection with inverse $\begin{array}{rcl} ℛ^{+} & \to & GL \\ β & \mapsto & l (β) \end{array}$ given by $l (β) = \max \{l (β_{1}) l (β_{2}) | β_{1}, β_{2} \in ℛ^{+}, β_{1} + β_{2} \in β, l (β_{1}) < l (β_{2})\}$
PBW basis and canonical basis of $U^{-}$
$U^{-}$ has basis $\{E_{g} | g \in G\},$ where $E_{g} = [l_{1}] [l_{2}] \dots [l_{k}] if g = l_{1} \dots l_{k} with l_{1}, \dots, l_{k} \in L and l_{1} \geq \dots \geq l_{k} .$
The ordering on $ℛ^{+}$ induced by lexicographic ordering on $GL$ determines a reduced ecomposition of $w_{0},$ $w_{0} = s_{i_{1}} \dots s_{i_{N}}, so that ℛ^{+} = \{β_{1} < \dots β_{N}\}$ with $β_{1} = α_{i_{1}}, β_{2} = s_{i_{1}} α_{i_{2}}, \dots, β_{N} = s_{i_{N}} α_{i_{N}} .$ Let $- : U^{-} \to U^{-}$ be the automorphism given by $\bar{f_{i}} = f_{i} and \bar{q} = q^{- 1} .$ The canonical basis of $U^{-}$ is $\{b_{g} | g \in G\}$ given by $\bar{b_{g}} = b_{g} and b_{g} = E_{g} + \sum_{h \in G, h > g} c_{g h} E_{h},$ with $c_{g h} \in q ℤ [q] .$

HW: If $+ < -$ then $G \cap L = \{+ - + -\}$ and $ℛ^{+} = \{α_{+}, α_{-}, - α_{+} + α_{-}\}$

Dual PBW and dual canonical bases

The dual PBW basis of $U^{-}$ is $\{E_{g}^{*} | g \in G\}$ given by $⟨E_{g}, E_{h}^{*}⟩ = δ_{g h} .$

(Lusztig) $E_{h}^{*} = (const) E_{h} .$
(Leclerc) $\{b_{g}^{*} | g \in G\}$ is characterised by $b_{g}^{*} = E_{g}^{*} + \sum_{h \in G, h \neq g} d_{g h} E_{h}^{*}, with d_{g h} \in q ℤ [q],$ and $b_{g}^{*} = \sum_{h \in I^{*}} a_{g h} h, with a_{g h} \in ℤ [q + q^{- 1}] .$ $U^{-}$ has bases $PBW basis:: \{E_{g} | g \in \vec{G}\},$ $dual PBW basis:: \{E_{g}^{*} | g \in \vec{G}\},$ $canonical basis:: \{b_{g} | g \in \vec{G}\},$ $dual canonical basis:: \{{b_{g}}^{*} | g \in \vec{G}\} .$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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Data for quiver Hecke algebras