What does the Leviathon represent? Why does Gavriel kill it? What does Gavriel represent that he should merit killing the Leviathon in the time to come? Where can we find more information about the Leviathon?
Sarah Brafman, Jerusalem, Israel
(a) The RASHBA in Peirushei ha'Hagados (Bava Basra 74a, p. 98 in Mosad Rav Kook edition) discusses at length the metaphorical meanings of the male and female Leviathan and Behemoth that will be devoured by the righteous in the World to Come.
He explains that the Leviathan refers to a person who uses his mind properly to discriminate between good and evil. When the Sechel, or the discriminating element of the mind (the male Leviathan), combines with the Nefesh, or the person's ability to think (the female Leviathan), in such a manner, he is referred to as "Leviathan." The word "Leviathan " implies a bond (or "Chibur"), as in the verse, "ha'Pa'am Yilaveh Ishi Elai" -- "Now my husband will become attached to me" (Bereishis 29:34). It also means an adornment worn upon the head, as in the verse, "Ki Livyas Chen Hem l'Roshecha" -- "for they shall be an adornment of grace for your head" (Mishlei 1:9).
(The male and female "Behemoth," as its name implies, represents the more mundane aspects of man: the material ("Chomer") from which the body is made, and the form ("Tzurah") that that material is given.) Based on these definitions, he proposes wonderfully enlightening explanations for numerous Agados in this Sugya.
With regard to Gavriel, the Rashba explains that Gavriel's killing of the Leviathan represents Hash-m's bringing an end to the physical existence that we know of (since the Leviathan represents intellect embodied in flesh, and that will come to an end). This is alluding to the Gemara in Avodah Zarah 9a which says that the world as we know it will come to an end after an allotted time span. Gavriel is the one who brings it to an end, since, as his name implies, he represents the Gevurah = Midas ha'Din (strict justice) of Hash-m. (In the World to Come, the world will be treated with Din.)
(b) However, the Mekubalim (SEFER HA'PELIAH, SEFER HA'TANYA #16, Rav Tzadok ha'Cohen in PRI TZADIK, Adar #2) write that the Leviathan represents the Yetzer ha'Ra (Evil Inclination) that is associated with lust, and Gavriel represents the Yetzer ha'Tov, "Eizehu Gibor ha'Kovesh Es Yitzro" (see also Rashi Sanhedrin 44b).
According to this view, when the Gemara says that Gavriel cannot conquer the Leviathan alone, it is a parallel to the Gemara in Kidushin 30b that says one would not be able to overcome the temptations of the Yetzer ha'Ra if not for the unique Divine support that Hash-m provides the Tzadik. When the Gemara discusses the slaughter of the Leviathan in Olam ha'Ba, it is a parallel to the Gemara in Sukah 52b that discusses how Hash-m will slaughter the Yetzer ha'Ra in the future.
(In truth, I believe the explanations of the Mekubalim and of the Rashba complement each other and are not mutually exclusive.)
I hope this is helpful for you,