 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-2/4/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Every ordered monomial in the degree-

n

homogeneous component has the form

x^{n - j} y^{j}

0 \leq j \leq n

. The formulas in part (a) show

K (x^{n - j} y^{j}) E (x^{n - j} y^{j}) F (x^{n - j} y^{j}) = q^{n - 2 j} x^{n - j} y^{j}, = [j] x^{n - j + 1} y^{j - 1}, = [n - j] x^{n - j - 1} y^{j + 1} .

Each nonzero image still has total degree

n

;

K^{- 1}

does too. Since

E, F, K, K^{- 1}

generate the quantum enveloping algebra of sl2, the degree- $n$ component is a submodule, with basis

x^{n}, x^{n - 1} y, \dots, y^{n}

n + 1

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