The quantum group as a shuffle algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 4 March 2012

The shuffle algebra and quantum group Uq𝔫-

Let be the free algebra generated by f1,, fn. Define a new product on by uv = σ Sk+l/ ( Sk×Sl ) qwt(σ,u,v) σ(uv) where k=l(u), =l(v), the sum is over all minimal length coset representatives of the cosets in Sk+l/ ( Sk×Sl ) and wt(σ,u,v) = 1i<jk+ σ(i)> σ(j) -ui, vj-k, where ui is the ithletter in the word u and vj-kis the (j-k)th letter in v. By [Le, prop. 1], is associative. Note that the powers of q in uv 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 =xbwa so that the product is the opposite of *.

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

Let a let a|u denote the coefficient of the word u in the expansion of a. Then a Uq𝔫- 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 ij with αi, αj 0 and all words z,t.

Example. (Type A2) If C= ( 2-1 -12 ), with α1 =α1 and α2 =α2, then Uq𝔫- contains 1,f1, f2, f1f2 = f1f2+qf2f1 , f2f1 = f2f1+qf1f2 and f1(f1f2) = ( 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 Uq𝔫- is the one which arises naturally as the character ring of the category of finite dimensional graded quiver Hecke algebra modules.


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

page history