Were Charlemagne and Ādi Shankara alive at the same time?
Yes. Charlemagne and Ādi Shankara were alive at the same time, from c. AD 788 to AD 814.
Charlemagne
King of Franks, regarded as the first Holy Roman Emperor (747–814)
Born AD 740–748 · died 28 January AD 814
Ādi Shankara
An Indian philosopher and theologian who consolidated the doctrine of Advaita Vedanta and revitalized Sanskrit scholasticism.
Born c. AD 788 · died AD 817–823