Hemispheric lateralisation is the idea that the two halves of the brain are functionally different and the certain mental processes and behaviours are mainly controlled by one hemisphere rather than the other.
split brain research is a series of studies which began in the 1960s involving people with epilepsy who had experienced a surgical separation of the hemispheres of their brain to reduce the severity of their epilepsy. this enabled researchers to test lateral functions of the brain in isolation.