Ringel duality as an instance of Koszul duality

Abstract

In [KKO14], S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra, i.e. the opposite algebra of the left dual, of a coring. Let A be an associative algebra and V an A-coring whose right algebra R is quasi-hereditary. In this paper, we give a combinatorial description of an associative algebra B and a B-coring W whose right algebra is the Ringel dual of R. We apply our results in small examples to obtain restrictions on the A ∞ -structure of the Ext-algebra of standard modules over a class of quasi-hereditary algebras related to birational morphisms of smooth surfaces. Date

From quasi‐hereditary algebras with exact Borel subalgebras to directed bocses

“…Example For the algebra in Example 3.2 the method given in [14] produces an exact Borel subalgebra of which is isomorphic to the path algebra of the quiver with relation . More examples of exact Borel subalgebras can be found in [4, 13, 14]. …”

All quasihereditary algebras with a regular exact Borel subalgebra

“…If (A, Φ, ✂) is a quasihereditary algebra, any refinement of the poset (Φ, ✂) gives rise to an equivalent quasihereditary algebra (see Lemmas 2.3 and 2.12 in [21], based on [17, pp. [3][4] and [10,Proposition 1.4.12]). One may want to remove relations from (Φ, ✂) in order to obtain a subposet (Φ, ✂ ′ ) of (Φ, ✂) with respect to which A has the same quasihereditary structure.…”

“…The bocses associated to quasihereditary algebras obtained by Koenig, Külshammer and Ovsienko in [29] are especially nice: they are directed. 6,29,30,4]). A bocs B = (B, W, µ, ε) is directed if the following conditions hold:…”

“…The next result was communicated to me by Julian Külshammer and is part of his joint work with Vanessa Miemietz. In fact, the statement of Lemma 3.7 is essentially an extension of Lemma 9.4 in [4] to the case where the underlying algebra of the bocs is not necessarily basic. Lemma 3.7.…”

“…(1) for every sequence of positive integers (k i ) i∈Φ there exists a quasihereditary al- By combining parts (1) and (2) of Theorem A, we conclude that the isomorphism classes of quasihereditary algebras in [(A, Φ, ✂)] having a regular exact Borel subalgebra are in one-to-one correspondence with the sequences of positive integers (k i ) i∈Φ . Furthermore, the representatives of [(A, Φ, ✂)] that contain a regular exact Borel subalgebra can be computed through the recipe in (2) and, by (4), there exists essentially one algebra in [(A, Φ, ✂)] that contains a basic regular exact Borel subalgebra. Theorem A therefore solves all the problems raised in (A).…”