Based on “Communicative and Cultural Memory,” by Jan Assmann, what is cultural memory and how does it apply to literature? How does the literary representation of historical events help create our sense of cultural identity? How can I approach this assignment?

Assmann distinguishes between two forms of communal memory, the shared memories of a group of people. Communicative memory goes back only about 80 years and is the group memory of events at least some living people can remember. For example, the bombing of Pearl Harbor is a communicative memory because a cohort of individuals still recalls it. The assassinations of John F. Kennedy and Martin Luther King and the destruction of the World Trade Center in 2001 would also qualify as living communicative memories.
Cultural memories are also shared, communal memories, but they go back, according to Assmann, as far as myths of origins: they are collective stories, supported by art objects, dance, and ritual, that shape the identity of a group. Elites control how these memories are shared and framed. Therefore cultural memories reflect the beliefs of those in power.
Assmann cites religious elites that safeguard and form cultural memories, such as the Brahmins in Hinduism, a high-ranking caste, and the Jewish Kabbala, an elite group of Jewish scholars over the age of 40 who safeguard esoteric (secret) knowledge. More broadly, however, cultural memories are the narratives, including those dating back before any living person's birth, that tell a story about a culture. You might approach cultural memory in the United States through stories of the first Thanksgiving. You could look into retellings of this story and its reinforcement through an annual ritual feast that transmits a sanitized myth of origin that tells a story both benign and multicultural (the English and the "Indians" get along and help one other); this communicates a picture of the United States as a white, Protestant, English nation.
You might also look at how culture is transmitted through literature such as the biblical account of the Jewish escape from Egypt in the book Exodus, as well as its embodiment in other cultural forms, such as Handel's oratorio Israel in Egypt. Cultural memories of the Holocaust reside in the works of Primo Levi and Victor Frankl, both camp survivors whose texts will continue to be influential past the 80-year mark for collective memory. Levi's Survival in Auschwitz is particularly interesting because he devotes a whole chapter to what we would now call cultural memory, describing how he, an Italian, regained a sense of humanity through reciting a canto from Dante's Inferno to another prisoner. Jane Austen's novels also function as artifacts of cultural memory because they are understood to transmit particularly English values. This led Kipling to valorize her as the emblem of civilization within the context of the barbarity of World War I in his short story "The Janeites" and has arguably led to her adoption as the face on the new British 10-pound note. 
Others who have done work on cultural memory include Susan Stewart and Pierre Nora.