The shuffle algebra and quantum group ${U}_{q}{𝔫}^{-}$

Let $ℱ$ be the free algebra generated by ${f}_{1},\dots ,{f}_{n}$. Define a new product on $ℱ$ by $u∘v = ∑σ∈ Sk+l/ ( Sk×Sl ) qwt(σ,u,v) σ(uv) where k=l(u), ℓ=l(v),$ the sum is over all minimal length coset representatives of the cosets in ${S}_{k+l}/\left({S}_{k}×{S}_{l}\right)$ and $wt(σ,u,v) = ∑ 1≤i σ(j) -⟨ui, vj-k⟩,$ where ${u}_{i}$ is the ${i}^{\mathrm{th}}$letter in the word $u$ and ${v}_{j-k}$is the ${\left(j-k\right)}^{\mathrm{th}}$ letter in $v$. By [Le, prop. 1], $\circ$ is associative. Note that the powers of $q$ in $u\circ v$ depend on the choice of the matrix of the form.

Remark. Leclerc [Le, §2.5 (8)] recursively defines a product by $wa*xb= (w*xb) a+q- ⟨deg(wa), deg(b)⟩ (wa*x)b,$ for words $w$ and $x$ and letters $a$ and $b$. By a straightforward check $wa*xb=xb\circ wa$ so that the product $\circ$ is the opposite of $*$.

The quantum group $Uq𝔫- is the subalgebra of (ℱ,∘) generated by f1,…, fr.$

Let $a\in ℱ$ let ${a|}_{u}$ denote the coefficient of the word $u$ in the expansion of $a$. Then $a\in {U}_{q}{𝔫}^{-}$ if and only if $∑k=0 -⟨ αi, αj∨⟩ +1 (-1)k [ 1-⟨ αi, αj∨⟩ k ] q2/⟨ αi, αi⟩ aTT | z fikfj fi1- ⟨αi, αj∨⟩ -k t=0,$ for all $i\ne j$ with $⟨{\alpha }_{i},{\alpha }_{j}^{\vee }⟩\ne 0$ and all words $z,t$.

Example. (Type ${A}_{2}$) If $C= ( 2-1 -12 ), with α1∨ =α1 and α2∨ =α2,$ then ${U}_{q}{𝔫}^{-}$ contains $1,{f}_{1},{f}_{2}$, $f1∘f2 = f1f2+qf2f1 , f2∘f1 = f2f1+qf1f2$ and $f1∘(f1∘f2) = ( f1f1f2 + q-2 f1f1 f2 + q-1 f1f2 f1) + q( f1f2f1 + q f2f1f1 + q-1 f2f1f1 ) = (1+ q-2) f1f1f2 + (q+q-1) f1f2f1 +(1+q2) f2f1f1 ,$ and the Serre relation for this element (Proposition ??? above) is $[ 1+1 0 ] (1+q-2) - [ 1+1 1 ] (q+q-1) + [ 1+1 2 ] (1+q2) = (1+q-2) - (q+q-1) (q+q-1) + (1+q2) =0.$

Notes and References

These notes are a presentation of the point of view of J.A. Green [Gr], which views the quantum group as a subalgebra of the free associative algebra instead of as a quotient. This version of the quantum group ${U}_{q}{𝔫}^{-}$ is the one which arises naturally as the character ring of the category of finite dimensional graded quiver Hecke algebra modules.

References

[Le] B. Leclerc, Dual canonical bases, quantum shuffles and $q$-characters, Math. Zeitschrift 246 (2004) 691-732 MR2045836 arXiv:math/0209133v3