# About the setting of the book “Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$”

I would like to ask about the setting of the book "Representations of Semisimple Lie Algebras in the BGG Category $$\mathcal{O}$$" by Humphreys. I would like to know whether the semisimple Lie algebra $$\mathfrak{g}$$ is finite dimensional or not. I search the book and did not find any assumption about the semisimple Lie algebra $$\mathfrak{g}$$ being finite dimensional. However, the book mentions root space decomposition of $$\mathfrak{g}$$, which depends on the finite dimensionality of $$\mathfrak{g}$$ by Humphreys' other book "Introduction to Lie Algebras and Representation Theory". Also a Cartan subalgebra $$\mathfrak{h}$$ of $$\mathfrak{g}$$ is mentioned to be finite dimensional on p.1 of "Representations of Semisimple Lie Algebras in the BGG Category $$\mathcal{O}$$".

My questions:

1. Is the semisimple Lie algebra $$\mathfrak{g}$$ finite dimensional?
2. If $$\mathfrak{g}$$ is not assumed to be finite dimensional, do we still have the root space decomposition of complex semisimple Lie algebra with respect to a Cartan subalgebra?
3. If $$\mathfrak{g}$$ is not assumed to be finite dimensional, does the root space decomposition imply that any complex semisimple Lie algebra with a Cartan subalgebra is finite dimensional?
4. If $$\mathfrak{g}$$ is not assumed to be finite dimensional, how do we prove $$\mathfrak{h}$$ is finite dimensional?
• Yes, the Lie algebra $\mathfrak{g}$ is finite dimensional. I'm sorry if this traditional setting isn't clear enough, but beyond this is mainly the Kac-Moody case which is not usually viewed as "semisimple". – Jim Humphreys May 8 '19 at 22:17
• Thank you for your reply. – James Cheung May 9 '19 at 6:42
• By the way, I would like to know does $\mathfrak{g}$ is complex semisimple Lie algebra imply $\mathfrak{g}$ is finite dimensional by the root space decomposition or do people just omit the words "finite dimensional" when assuming $\mathfrak{g}$ is finite dimensional complex semisimple since it is a traditional setting? – James Cheung May 9 '19 at 6:51

1. In the book, $$\mathfrak{g}$$ is always assumed to be finite-dimensional. (I believe that this is explicitly stated in the first chapter of the book.)
2. Depending on $$\mathfrak{g}$$ and on the definition of Cartan subalgebras in the infinite-dimensional setting, you may have an infinite-dimensional semisimple Lie algebra $$\mathfrak{g}$$ and a Cartan subalgebra $$\mathfrak{h}$$ such that, with respect to $$\mathfrak{h}$$, the root-space decomposition of $$\mathfrak{g}$$ exists. You can look at something like root-reductive Lie algebras.
• Thank you for your answers. But I really cannot find out the assumption $\mathfrak{g}$ is finite dimensional throughout the book by manually searching the word "finite". I just find out some sentences implicitly suggest that $\mathfrak{g}$ is finite dimensional. e.g. p.36 (the last sentence) and p.259 (the last sentence in Section 13.6). If you can find it out, please let me know. – James Cheung May 8 '19 at 12:15