62 Algebraic Operads: An Algorithmic Companion
(i) Using the glex order on T (x, y) for, say, x ≺ y, compute the reduced
Gröbner basis for the principal ideal (x
2
− xy + y
2
).
(ii) Compute the center of the algebra A = T (x, y)/(x
2
−xy + y
2
), and find
the structure of A as a module over its center.
(iii) Classify finite-dimensional complex irreducible A-modules, and solve the
problem stated above.
Exercise 2.7. Pick a monomial order of T (x
11
, x
12
, x
21
, x
22
), and compute
the reduced Gröbner basis for
(i) the algebra of quantum 2 × 2-matrices,
(ii) the quantum group SL
2
(see Example 2.1.1.4 for definitions). Verify that the dimensions of homoge-
neous components of these algebras over F(q) are the same as the dimensions
over F for the algebras obtained by setting ...