first, Brahman
I would recommend you first read this answer on Hindusim SE - What is the etymology of brahman/brahmin?
In his Sri-Bhasya to the Brahma Sutras (I. I. 1.), Ramanujacharya says (Swami Vireswarananda translator):
The word ‘Brahman’ is derived from the root ‘brh’ which denotes greatness, and is therefore applicable to all objects which have the quality of greatness, but more aptly to that object which by nature and by qualities possesses this greatness to an infinite degree, hence the word ‘Brahman’ denotes that supreme Person who is the abode of all auspicious qualities to an infinite degree and is free from all worldly taint.
And in his book Bhamati and Vivarana Schools of Advaita Vedanta: A Critical Approach, p 190, P.S. Roodurmun says:
As for the meaning of the word ‘Brahman’ itself, Padmapada says [in his book Pancapadika] that with reference to the root Brh (to be great) from which is derived, the word Brahman is denotative of an entity possessed of unexcelled magnitude, which meaning alone fits in with other appellations namely, Absolute Existence (Satyam), Consciousness, (Jnanam), Infinite (Anantam), as per the Vedanta text—“Satyam Jnanam Anantam Brahma”. Hence, says Padmapada, because of the absence of sense of limitedness occasioned by the delimitation of time, some entity that is constant and therefore eternal is intimated by the term Brahman…
[p 192]…The author of of the Pancapadika says concludingly that the meaning of the root ‘Brh” will be complete only if the whole universe lies within the scope of (Brahman’s) Consciousness or omniscience, and is under Its control, as Sankara himself says: "There verily exists an entity referred to as Brahman having Eternity, Purity, Consciousness, Freedom, Omniscience, and Omnipotence as its characteristics, as is revealed by the root ‘Brh' itself".
Swami Vivekananda says that when the term Maya first appears in the Rig Veda, it is used in what can be translated as ‘magic’ or ‘illusion’; or as ‘magic force’ and only later in the Upanishads evolved into ‘cosmic force’; but nowhere is the meaning of Brahman associated with ‘magic’.
ATMAN
Swami Vivekananda said in a lecture (Complete Works, V3, p 401, available here under the heading Lectures from Colombo to Almora, sub-heading The Vedanta - http://cwsv.belurmath.org/volume_3/vol_3_frame.htm):
I will not translate this word [Âtman] to you in English, because the idea does not exist in Europe; it is untranslatable. The modern attempt of German philosophers is to translate the word Atman by the word "Self", and until that word is universally accepted, it is impossible to use it. So, call it as Self or anything, it is our Atman.
The translation of Atman as ‘Self’ or ‘self’ (there is no universal agreement as to the capitalization) is at best a signifier for the word Atman when rendered into English, and should not be taken as a literal translation or universally accepted translation.
Again in Bhamati and Vivarana Schools of Advaita Vedanta: A Critical Approach, p 191-192, P.S. Roodurmun says:
Sankara, by asserting the existence of the Self (Atman) and by identifying it with Brahman, proves the existence of the Supreme Being. This, he says, is very much within the experience of one and all or else, instead of cognition “I am”, everyone would have the cognition “I am not”, which is absurd. Reacting to this, Padmapada says that Atman is identifiable with Brahman on account of the assertion by the Vedanta texts in this sense. And this Atman is indeed the entity designated by the ego (aham) in this empirical world. Hence, the ego itself is Brahman. Clarifying this point further, the author of the Pancapadika says that ‘the word aham means Consciousness divested of all limiting adjuncts’.
Thus Atman is Brahman, and is never the 'human person' meant. The innermost soul of man is the atman, and the atman is the same as Brahman. For further references, see Mundukya Upanishad I. 2. And Sankara’s commentary on it. This verse contains the Mahavakya – “ayam atma brahma" – This Self is Brahman.
Carvaka (Lokayatas)
In his book The Spiritual Heritage of India, Swami Prabhavananda says (p 17):
The systems of Indian philosophy fall into two main divisions according as they do or do not accept the authority of the Vedas. All systems except Buddhism and Jainism are pronounced astika—meaning, in effect, orthodox; these two, which deny the authority of the primary scriptures, are nastika—unorthodox. If, however, we interpret astika literally—belief in existence after death—then all systems of thought, with the exception of the system attributed to Carvaka are astika.
What Carvaka really taught, or whether there was a philosopher named Carvika at all, it is difficult to know, for we hear of him only through refutation, by various other schools of thought, of a philosophy of sensualism attributed to him. This philosophy was, in effect, but the simple philosophy of skepticism which appears as a crosscurrent in every age and every country. The name Carvaka literally means sweet word.
Some oriental scholars translate nastika as atheist. But if this meaning of the word is applied to Buddhism and Jainism because they reject an anthropomorphic God, then it should be applied also to many of the orthodox schools. The Samkhya philosophy, for example, denies God as creator, yet it is held to be orthodox.
Curiously, there is no equivalent in Sanskrit for the word atheism. In the Gita mention is made of those who do not believe in God, the intelligent principle, but these are spoken of merely as of ‘deluded intellect.’
And in his introduction to his translation of the Brahma Sutras According to Sri Sankara (available here - http://www.wisdomlib.org/hinduism/book/brahma-sutras/d/doc62753.html) Swami Vireswarananda says (pp ii-iii):
…Traces of opposition against the religion of the Vedas are found in the Vedas themselves. This tidal wave of rationalism in its extreme form gave rise to to such schools of thought as the Carvakas, which were extremely materialistic and anti-religious.
In the age immediately preceding Buddha and during his lifetime there was a great religious and philosophical upheaval in India. From the Brahma-jala-Sutras we learn that in his time there were as many as sixty-two different schools of philosophy in India. We also learn from Buddhistic literature the names of number of teachers who were venerated in Aryavarta at the time—like Purana Kasyapa, Katyayana, Makkali Gosala, Nigantha Nathaputra the founder of Jainism and others. While these great souls represented Indian culture from an anti-Vedic standpoint there were many great names that represented the culture from the traditional standpoint—names that are venerated by Hindu religion and culture.
The destructive criticism of everything in the old system by the Carvakas and others set the orthodox section to organize their belief on a more rationalistic basis and render it immune against all such criticism. This led to the foundation of the six systems of orthodox Hindu philosophy—orthodox [footnote – Astika (orthodox) and Nasika (heterodox) had nothing to do with belief or non-belief in the existence of God. Samkhya and Mimamsa which did not accept an Iswara were yet regarded Astika (orthodox)] in the sense they accepted the authority of the Vedas in things transcendental—while there were others who did not accept this authority and therefore were dubbed heterodox, though otherwise they too were the outcome of Upanisadic thought. The acceptance of the authority of the Vedas by these orthodox schools, however, does not mean that they accepted them in toto. Their allegiance to the Vedas varied widely and often was too loose. Of the six orthodox schools, viz Nyaya, Vaisesika, Samkhya, Yoga, Purva Mimamsa and Uttara Mimamsa or Vedanta [all modern Hindus], the last two are intimately connected with the Vedas, which is one of the reasons they are not mentioned in the Jaina and Buddhistic literature, while the others are mentioned.
If you read the Brahma Sutras with either Sankara’s or Ramunuja’s commentary (link for Sankara’s above), arguments are made in various verses against the Carvakas views. But as pointed out above, no literature of theirs has survived.
