Gemara

Eliyahu (to R. Yosi): When Yisre'elim enter Batei Kenesiyos and Batei Medrash and proclaim "Yehei Shmei ha'Gadol Mevarach," Hash-m nods His head and says "how fortunate is the King Whom they praise in His house like that! Woe to the Father Who exiled His children, and woe to the children who have been exiled from their Father's table!"

Shabbos 119b (R. Yehoshua ben Levi): If one answers 'Amen Yehei Shmei Raba' will all his Ko'ach (concentration), harsh decrees against him are torn up

Sukah 39a (Rava): One should not say "Yehei Shmei Raba" and later Mevarach. Rather, he says "Yehei Shmei Raba Mevarach" together;

Rebuttal (Rav Safra): "Mevarach" concludes his words. There is no problem (if he pauses before it).

Rishonim

Rambam (Seder Tefilos (b'Sof Hilchos Tefilah), Nusach ha'Kaddish): When the Shali'ach Tzibur says v'Imru Amen, everyone says Amen Yehei Shmei Raba Mevarach l'Alam ul'Almei Almaya.

Hagahos Ashri (3:9): One should not interrupt between "Shmei" and "Raba". One may interrupt between Raba and Mevarach.

Tosfos (3a DH v'Onin): Machzor Vitri (of Rashi) says that Yehei Shmei Raba is a prayer that Hash-m's name be complete, like it says "Ki Yad Al Kes Kah" - Hash-m's Name and throne will not be complete until He wipes out the seed of Amalek. The Perush is Yehei Shmei, i.e. the Shem (name) Yud Kei should be Raba. I.e. we pray that His Name should be great and complete. Mevarach l'Olam is a different prayer, that it should be blessed in the world to come. Our Gemara refutes this. It is one request. We do not request that His Name be great and complete. Rather, His great Name should be blessed. People say that Kaddish is in Arame'ic for it is a beautiful praise, so we do not want the angels to understand it and be envious. This is wrong. There are many beautiful praises in Lashon ha'Kodesh! Rather, in Sotah (49a) we say that the world stands on Kedushah d'Sidra (in u'Va l'Tziyon) and Yehei Shmei Raba after Agadata. They used to say Kaddish after the Drashah, and ignoramuses did not understand Lashon ha'Kodesh, so they enacted it in Arame'ic, so all would understand it.

Tosfos (Sukah 38b DH Amar): In Yevamos (106b), Rava is not concerned for a break between "Lo" and "Ava Yabmi", for the latter concludes her words! After he heard from Rav Safra, he retracted.

Poskim

Rema (OC 56:1): One should not interrupt between "Yehei Shmei Raba" and Mevarach. (Hagahos Ashri says that the first Perush says not to interrupt between Shmei and Raba, and the Ri says not to interrupt between Raba and Mevarach.)

Taz (1): According to the Perush that Kaddish is in Arame'ic so that people will understand, Shmei Raba is the translation of "His great Name". The other Perush cannot explain simply like this, for a request that His Name be blessed would not arouse such envy.

Magen Avraham (2): The Shach brings this (not to interrupt) from Darchei Moshe and the Bach. This is difficult, for in Sukah we say that there is no problem! The Rema holds that Hagahos Ashri holds like the first Perush (in Tosfos 3b), that Yehei Shmei Raba is a request that Hash-m's, which is now only half, be completed. Mevarach is a different request. The Rema inferred that the Ri holds that we request that His great Name be blessed. If so, one should not interrupt even between Raba and Mevarach. This requires investigation, for it is unlike the Gemara! Granted, Hagahos Ashri can say that Rav Safra permits a Hefsek only between Raba and Mevarach. for it is not one statement, but not between Shmei and Raba. However, the Ri forbids interrupting in either place! I say that the Rema learns from EH 169:29. L'Chatchilah, a Yevamah should read in one breath. Also here, l'Chatchilah one should not interrupt. One need not say it in one breath. According to the Ri, one says Shmei without a Mapik Hei (for it means His name).

Gra (DH v'Lo): Tosfos holds that Kaddish is in Arame'ic so that all will understand it. Also Shmei is Arame'ic (His name). We request that His great Name be blessed, therefore one may not interrupt between Raba and Mevarach. The Beis Yosef holds like this. Therefore, he says (below, DH Kasav) that there is no Yud in Shmei. According to the opinion that Kaddish is in Arame'ic so the angels will not understand it, Shmei is really Hebrew (Shem Yud Kei). Since they do not understand the rest, we are not concerned if they understand this word. We request that His Name be complete, therefore one may not interrupt between Shmei and Raba. It is difficult to resolve this with Sukah 39a. The Magen Avraham's answer is wrong. The Darchei Moshe learned from Hagahos Ashri, who wrote like the Gemara there! Since he concludes his words, we are not concerned for a Hefsek. The same applies between Shmei and Raba!

Chasam Sofer: I say that one may pause silently. The Rema says not to put in between a Ta'am (note, e.g. inflection of voice) that designates a stop.

Mishnah Berurah (3): He should not pause in between, but he need not say them in one breath.

Kaf ha'Chayim (11): The Bach says that one should not pause at all, in order to fulfill both Perushim. The Ari Zal says that the primary Perush is Rashi's. There are 11 letters in Yisgadel v'Yiskadesh. They allude to the 11 ingredients in the Ketores, and the Gematriya of the last two letters of Hash-m's name. During the Galus, Klipos (shells of Tum'ah) cling to these two letters, so they are not properly attached to the first two letters.

Shulchan Aruch (3): Those who answer only until "l'Almei Almaya" err. One may not separate between Almaya and Yisbarach.

Beis Yosef (DH Kasav, citing Avudraham): Some people answer Amen Yehei Shmei Raba... until v'Imru Amen. I received that one answers only until "ul'Alam Almaya". Rav Sadya Gaon connotes like this, and so it is written in the name of the Rosh. Yehei Shmei Raba has seven words and 28 letters without Amen, which is separate from it. Also, this is from the verse (Daniel 7:8) "...v'Ad Alam Almaya."

Beis Yosef (ibid.): Also the Kolbo says so. He says not to say ul'Alamei Almaya, lest there be 29 letters. Rather, we say v'Almei Almaya. Mahari Avuhav says to say l'Alamei Almaya without a Vov, so there will be 28 letters. I say that we do not change texts that Chachamim made due to any Drashah. We received to say ul'Alamei Almaya. Some defend this, and say that Shmei has no Yud, like in Daniel 2:20.

Mishbetzos Zahav (3): In Daniel, Shmei, the Targum of His Name, has no Yud and a Mapik Hei. (Therefore, according to both Perushim one should say it with Mapik Hei.)

Beis Yosef (DH ul'Inyan): Also Rashi's Sefer ha'Pardes and the Rambam say to answer only until Almaya. Mahari Avuhav says that some do not require answering past Almaya, so the response will have seven words and 28 letters, like the first verse "Bereishis..." and the verse before the 10 utterances. A hint is 'one who answers with Kol Kocho'. (The Gematriya of Ko'ach is 28.) Some say that one must answer 28 words, until d'Amrin b'Alma, to fulfill "Ko'ach Ma'asav Higid l'Amo." A Sidur of Ge'onim permits answering only until Almaya, but if he says Yisbarach, he must complete the Kaddish and not answer Amen. The Shali'ach Tzibur says "v'Imru Amen" only to those who did not say with him. Since opinions differ about how much to answer, when one interrupts Keri'as Shema and its Berachos to answer, he answers only until Almaya, for perhaps past that is a Hefsek. If not, he says until d'Amrin b'Alma and says Amen.

Magen Avraham (9): The Beis Yosef connotes that one must say 28 words until d'Amrin b'Alma. I say that the ancient custom is primary, to say only until Yisbarach. The Beis Yosef (Reish Siman 125) brought from the Rosh that one must be quiet and intend for the Kaddish that the Shali'ach Tzibur says. He should not say it, for we hold that one who hears is like one who says. Especially when the Shali'ach Tzibur rushes, the listener will need to rush even more in order to finish before the Shali'ach Tzibur. Rushing impedes Kavanah. Therefore, it is better to listen to the Shali'ach Tzibur.

R. Akiva Eiger: See the Targum Yerushalmi on Bereishis 49:1 and Devarim 6:4. (It says that when Yakov suspected lest one of his son is unworthy, they said Shema Yisrael..., and he responded Yehei Shmei Raba Mevarach l'Almei Almin (in Devarim, "...Mevarach la'Alam").

Beis Yosef (DH v'Kasha): I question those who say only until Almaya. A Medrash mentions a great punishment for one who talks between Yehei Shmei Raba and Yisbarach. This connotes one must say Yisbarach. It is difficult to say that the Medrash discusses the Shali'ach Tzibur. Perhaps an individual need not say Yisbarach, but he must hear the entire Kaddish from the Shali'ach Tzibur. If the Shali'ach Tzibur says Amen Yehei Shmei Raba at length, even if the individual already finished answering, he may not converse before the Shali'ach Tzibur says Yisbarach. R. Yosef Gikataliya said 'woe to the fools who separate between Almaya and Yisbarach. It is as if they cut where one may not cut. One must connect these words!'

Gra: The Mechaber learned from the Medrash that one must answer also Yisbarach. R. Yosef Gikataliya supported this from Kabalah. All the Poskim wrote to answer only until Almaya, so there will be 28 letters. The Ge'onim and Rambam agree. Really, the Medrash supports them. It considers between Almaya and Yisbarach like between Perakim (segments), like between Kaddish and Borchu. It forbids talking only between them, for an individual answers until there, and from there he listens quietly! Commoners often talk then. It did not discuss talking between Yehei Shmei Raba and Almaya, or during a Berachah (for this is not common). There is no proof from the Kabalah. They are different segments, but they must be adjacent. Separating them is the ultimate cutting! The Rambam's opinion is primary.

Mishnah Berurah (15): If one says Almaya and Yisbarach in separate breaths, also the Gra permits. There are different customs about this. In any case, in a place where one may not interrupt to talk, he should say only until Almaya.

Kaf ha'Chayim (33): The Mechaber rules like R. Yosef Gikataliya, that one may not separate Almaya from Yibarach. He did not reveal how much one must answer. The Magen Avraham and Eliyahu Rabah say that he requires answering until d'Amrin b'Alma (28 words). The Ari Zal says so. We totally follow him, even in the middle of Berachos Keri'as Shema. Whenever something is not explicit in the Gemara and Poskim argue, Kabalah decides the matter.