Later came Hillel and Shammai, both of whom were from the root of Abel. However, Hillel was from the side of chesed of Abel, and Shammai was from the side of gevura of Abel. Therefore G‑d says to Moses, "You need to help me", (Shabbat 89a) the head-letters of which spell Hillel. Indeed, both Moses and Hillel were known for their humility,1 since they were both from the side of chesed of Abel.

...both Moses and Hillel were known for their humility...

As well, they are both alluded to in the seventy-two letter name that comes from the verses "And he traveled," "And He went," and "And he set up".2 (Ex. 14:19-21) The two names within it, when close together, are the letters MHSh and LLH, which are the letters MShH-HLL/Moses-Hillel.

Also, Moses and Hillel were both kings. (Deut. 33:5 and Pirkei Avot) Each lived to the ripe old age of a hundred and twenty, and their life-spans were divided into three forty year periods.3 (Deut. 34:7, Mid.R.Shem., Rosh HaShanah)

"A forty year old for bina." (Pirkei Avot) Each sefira has its own right/left/middle aspect. In bina they each are forty by nature. One who draws from there merits the three forty year sections. From this comes the concept of the ideal of living to a hundred and twenty.

In order to step above the world, one must step out of himself. The quality of humility is so central to achieving these levels. This all links with the secret of the seventy-two letter name, a means to do sublime miracles with. It was engraved on Moses’ staff that he spread out over the sea to split it. Accordingly, Moses brought the Torah - the connection to above, to the world. Hillel was the central giver over of the Torah for generations.4 They are both inscribed in the seventy-letter names, empowering their souls to do such tasks.

...Shammai...was from the side of gevura of Abel.

However, Shammai, as mentioned, was from the side of gevura of Abel. So the gematria of Hillel is the same as ADNY/Adona"i, which is considered to be the panim [face/front (chesed-oriented aspect)] of malchut. Shammai was the achorayim [back (din-oriented aspect)] - which is why he was so strict, since he was from the side of gevura.

Thus, ShMAY/Shammai has two letters ShM from MShH/Moshe/Moses, which are also from ShM/Shem son of Noah. The two letters AY (of ShMAY) are the secret of (what God said to Cain) "AY [where is] Abel your brother." (Gen. 4:9) Since Shammai is from the gevurot of Abel, there is an allusion to the sin of Abel in his name. For he (Abel) transgressed with respect to the two letters AY from ADNY/Adona " i - hinted to by "AY [where] is Abel your brother," as mentioned in Tikunim, Tikun 69.

This can only properly be understood in light of the step before. After they ate from the tree, G‑d said to Adam, "AYKH/Ayeka [where are you]?! " (Gen. 3:9) The Zohar explains this to be, ‘AY-KH/ayeh-ko [where is ko]. 'Ko' is malchut.5 Meaning to say, "To where have you caused malchut to fall through your sin, and where are you as a result of it?"

Cain came from the evil that was injected into them by the snake. (Zohar) G‑d said to him only AY/ai [where], for the letters KV/ko were already gone. Instead of fixing his father’s mistake, he made it worse. Like Cain, Shammai is from the left side. He was therefore given these letters of Cain to repair.

The secret is just the depths of the simple explanation. (Vilna Ga’on on Birkat Hatorah) Cain was apathetic to his brother’s existence. Even after murdering him he said, "Am I my brother’s keeper?" (Gen. 4:9) G‑d hinted to him his mistake, and therefore his pipeline of repentance, by starting the conversation with, "Where is Abel your brother". As if to say, "Shouldn’t you care enough to know?"

...part of Shammai’s tikun was to care about those that came to him seeking the truth.

From this piece we see that apparently part of Shammai’s tikun was to care about those that came to him seeking the truth.

AY are the first and last letters of ADNY, the Divine name for malchut. As mentioned, Hillel comes from there. It is called the asplecraria she’aina m’ira [i.e. unshining looking-glass] - through which all of the prophets prophesized.6 One of the central ways to merit it is through caring about others. And so Hillel was known to say, "Don’t do to others what is hateful to you". (Shabbat)

A/aleph is the first letter of the alphabet and it includes them all. Y/yud is but a dot, and it is the only letter that does not reach the bottom of the line. In the official writing of the ashurit script that the Torah was given, a scribe starts to write any letter from the point of yud. Thus yud is also a letter of beginning.

This is the idea behind the AY, first and last letter of ADNY the name of malchut: The potential to connect finish to start. Together they equal eleven - the amount of spices in the ketoret [incense offered in the Temple]. The Zohar tells that there is nothing that raises the fallen sparks out of the kelipot like ketoret. Hillel the prince, who fished souls out of the sitra achra and brought them close to their Maker,7 rectified these letters.

[Translation and commentary by Perets Auerbach.]