Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 1 / b / i

Table of contents
- Solution i

Solution

The relation

[h, e] = 2 e

e w_{i}

to have weight

a + 2 (i + 1)

, so

e w_{i} = c_{i} w_{i + 1}

c_{i}

. The relation

[e, f] = h

becomes

(c_{i - 1} - c_{i}) w_{i} = (a + 2 i) w_{i} .

With

c_{0} = b

, this recurrence has the unique solution

c_{i} = b - ia - i (i + 1)

for every

i \in Z

. Direct substitution also verifies

[h, f] = - 2 f

and

[h, e] = 2 e

, so these formulas define the unique required sl2 Lie algebra action. They form an Intermediate-series sl2 module.

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