The quantum group as a shuffle algebra
Last updates: 4 March 2012
The shuffle algebra and quantum group
Let be the free algebra generated by
. Define a new product on by
the sum is over all minimal length coset representatives of the cosets in
where is the
letter in the word
letter in . By [Le, prop. 1], is associative.
Note that the powers of in
depend on the choice of the matrix of the form.
Remark. Leclerc [Le, §2.5 (8)] recursively defines a product by
for words and and letters
and . By a straightforward check
so that the product
is the opposite of .
The quantum group
denote the coefficient of the word in the expansion of
. Then if and only if
for all with
and all words .
Example. (Type )
and the Serre relation for this element (Proposition ??? above) is
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 is the one which arises naturally as the character ring of the
category of finite dimensional graded quiver Hecke algebra modules.
B. Leclerc, Dual canonical bases, quantum shuffles and -characters,
Math. Zeitschrift 246 (2004) 691-732