This grammar question, as in the case of many others, is most efficiently answered by adhering to a single style guide. As Dale Emery suggested, the Chicago Manual of Style is pretty awesome for finding answers. (I, too, am biased. Keep in mind that these are American rules of grammar and punctuation, though.)
TL;DR:
- The book by author states, "The cake is not a lie." However, studies show that the cake is a lie.
- The book by author states, "The cake is not a lie"; however, studies show that the cake is a lie.
- The book by author states, "The cake is not a lie," but studies show that the cake is a lie. (CMoS punctuation here)
- The book by author states, 'The cake is not a lie', but studies show that the cake is a lie. (British style here)
Here are relevant sections from the sixteenth edition of the Chicago Manual of Style:
6.9 PERIODS AND COMMAS IN RELATION TO CLOSING QUOTATION MARKS
Periods and commas precede closing quotation marks, whether double or single. (An apostrophe at the end of a word should never be confused with a closing single quotation mark; see 6.115.) This is a traditional style, in use well before the first edition of this manual (1906). For an exception, see 7.75.
See also table 6.1.
Growing up, we always preferred to "bear those ills we have."
"Thus conscience does make cowards of us all," she replied.
In an alternative system, sometimes called British style (as described in The Oxford Style Manual; see bibliog. 1.1), single quotation marks are used, and only those punctuation points that appeared in the original material should be included within the quotation marks; all others follow the closing quotation marks. (Exceptions to the rule are widespread: periods, for example, are routinely placed inside any quotation that begins with a capital letter and forms a grammatically complete sentence.) Double quotation marks are reserved for quotations within quotations. This system or a variation may be appropriate in some works of textual criticism. See also 7.50, 7.55, 13.7--8, 13.28--29.
6.10 OTHER PUNCTUATION IN RELATION TO CLOSING QUOTATION MARKS
Colons and semicolons--unlike periods and commas--follow closing quotation marks; question marks and exclamation points follow closing quotation marks unless they belong within the quoted matter. (This rule applies the logic that is often absent from the US style described in 6.9.) See also table 6.1.
Take, for example, the first line of "To a Skylark": "Hail to thee, blithe spirit!"
I was invited to recite the lyrics to "Sympathy for the Devil"; instead I read from the Op-Ed page of the New York Times.
Which of Shakespeare's characters said, "All the world's a stage"?
but
"Timber!"
"What's the rush?" she wondered.
and
6.50 COMMAS WITH QUOTATIONS
Material quoted in the form of dialogue or from text is traditionally introduced with a comma (but see 6.63, 13.17). If a quotation is introduced by that, whether, or a similar conjunction, no comma is needed.
It was Thoreau who wrote, "One generation abandons the enterprises of another like stranded vessels."
She replied, "I hope you aren't referring to us."
Was it Stevenson who said that "the cruelest lies are often told in silence"?
He is now wondering whether "to hold, as 'twere, the mirror up to nature."
For the location of a comma in relation to closing quotation marks, see 6.9. For quoted maxims and proverbs, see 6.51. For detailed discussion and illustration of the use or omission of commas before and after quoted material, including dialogue, see 13.18, 13.48--56, and the examples throughout chapter 13.
I'd also point to a dictionary's definition of "however." "However" is a conjunction only when it means "in whatever way," but it is an adverb when it denotes contrariness. A grammar teacher would call something like the original sentence a hidden comma splice; it is two independent clauses disguised as a single sentence.
So here are some thoughts based on these sections:
Firstly, when you introduce a quotation with "that," you are incorporating the quotation as your own words. This would call for a lowercase introduction (e.g., The author states that "the cake is a lie.") If you introduce the words as the author's words, then it is customary to include an introductory comma (e.g., The author states, "The cake is a lie.")
In American grammar, a period at the end of a quotation that does not end a sentence is customarily replaced with a comma within the quotation marks; in British style, the punctuation is outside of the quotation marks. In your case, it will probably be followed by a conjunction and an independent clause, which is all the more reason to include a comma somewhere. (Independent clauses joined by conjunctions generally have a comma separating them unless the clauses are very short or simple.)
Additionally, I mentioned my qualms with "however" used as a conjunction, and I'm sure any prescriptivist grammarian would agree. If you want, instead of starting a new sentence, you could rewrite it with a semicolon, as in: The book by author states, "The cake is not a lie"; however, studies show that the cake is a lie.
Lastly, I just want to mention something style-related. If you have a quotation and then a direct challenge or opposition to that quotation, especially one in which you repeat the phrasing, it's common to emphasize that difference in some way (e.g. "However, studies show that the cake is a lie," or "However, studies show that the cake is indeed a lie."). Just a style thing, though. :)
Commas are tricky things, and they are never so tricky as when they interact with quotation marks. I hope these sections and my analysis of your sentence have been at least a little illuminating.
