Tilting Objects in BGG Categories $\mathcal{O}$

Question

Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by Humphreys). If $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$ contained in $\mathfrak{b}$, then there exists a unique indecomposable module $D(\lambda)$, where $\lambda\in\mathfrak{h}^*$, such that the weight space of $D(\lambda)$ with weight $\lambda$ has dimension $1$, and $D(\lambda)$ has a finite filtration with standard subquotients as well as a finite filtration with costandard subquotients. The construction of $D(\lambda)$ is discussed in the same book.

However, when I read Ringel's definition of tilting modules, I noticed some discrepancies. For example, a tilting module $D(\lambda)$ does not need to have projective dimension $1$ in $\mathcal{O}$ (this is the first condition in the link). Plus, the enveloping algebra $U(\mathfrak{g})$ is not in $\mathcal{O}$, so it is not the kernel of any epimorphism from a direct sum of finitely many copies of $D(\lambda)$ to another direct sum of finitely many copies of $D(\lambda)$, violating the third condition in the link. (I am not sure anyhow if the third condition makes any sense. After all, $\mathcal{O}$ is not the whole category of $U(\mathfrak{g})$-modules.) The only condition $D(\lambda)$ satisfies is $\operatorname{Ext}_{\mathcal{O}}^1\big(D(\lambda),D(\lambda)\big)=0$ (i.e., the second condition of the link).

The first condition can be fixed by using the definition of generalized tilting modules, but this definition may be not-so-helpful for other abelian categories. However, there doesn't seem to be a fix for the third condition. There is still no exact sequence $0\to U(\mathfrak{g})\to T_0\to T_1\to T_2 \to \ldots \to T_n \to 0$, where each $T_i$ is a finite direct sum of $D(\lambda)$.

Hence, I would like to ask why are $D(\lambda)$ called tilting modules anyway? Is there a version of tilting theory that works with $D(\lambda)$? Do we have something like Ringel duality here? Is there a tilting theory for abelian categories in general?

For tilting objects in a general abelian category, I am aware of the work by Colpi and Fuller in 2007, but it requires that the category contains the direct sums of arbitrary, possibly infinite, numbers of copies of some objects. I think this requirement is too restrictive.

Answer 1

Words change their meanings.

The original meaning of “tilting module” is that of Happel and Ringel in the representation theory of finite dimensional algebras, which requires the projective dimension to be one, and a “generating” condition (the third condition in your link).

Fairly soon, it was recognized that generalizing from projective dimension one to finite projective dimension was worthwhile. Originally, this generalization was called a “generalized tilting module”, but nowadays it would be confusing not to specify projective dimension one if that’s what you mean.

Then people from the algebraic groups/Lie algebras community got interested in axiomatizing highest weight categories, and Ringel proved the existence of the objects $D(\lambda)$ that you describe in the first paragraph of your question, and that, in the case where the highest weight category is the module category of a finite dimensional algebra, that the direct sum of the modules $D(\lambda)$ is a tilting module (in the generalized sense).

The algebraic groups/Lie algebras people were apparently not paying enough attention, and started referring to the $D(\lambda)$ as “the tilting modules”, which is wrong on possibly as many as three counts.

But the two communities, finite dimensional algebras on the one hand and algebraic groups/Lie algebras on the other, are still on cordial terms, and gently mock each other about the use of the term “tilting module”, in the same way that Britons and Americans gently mock each other about the use of the word “pavement”.

(Of course, it’s the finite dimensional algebra people and the Brits who are correct, but live and let live, eh?)