In the bocs seat: Quasi-hereditary algebras and representation type

Külshammer, Julian

doi:10.4171/171-1/14

Cited by 10 publications

(13 citation statements)

References 54 publications

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“…The unique solution of the linear system consequently implies that every quasihereditary algebra with an indexing poset of height at most 2 has a regular exact Borel subalgebra. In particular, every quasihereditary algebra with at most two nonisomorphic simple modules has an exact Borel subalgebra: this had already been proved in [30,Theorem 4.58].…”

Section: ]) There Is a One-to-one Correspondence Between Regular Dire...mentioning

confidence: 80%

“…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:…”

Section: How Bocses Relate To Quasihereditary Algebras and Exact Bore...mentioning

confidence: 99%

“…As announced in [30,Theorem 4.26], a bijection between the isomorphism classes of regular directed bocses B = (B, W, µ, ε) with B basic and the equivalence classes of quasihereditary algebras, mapping a bocs B to the equivalence class of its right algebra R B , shall be provided in upcoming work of Külshammer and Miemietz. We have not been able to prove that every two regular directed bocses B = (B, W, µ, ε) and B ′ = (B ′ , W ′ , µ ′ , ε ′ ) with B and B ′ basic and R B and R B ′ equivalent must be isomorphic.…”

Section: Introductionmentioning

confidence: 99%

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All quasihereditary algebras with a regular exact Borel subalgebra

Conde

2020

Preprint

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Not every quasihereditary algebra (A, Φ, ✂) has an exact Borel subalgebra. A theorem by Koenig, Külshammer and Ovsienko asserts that there always exists a quasihereditary algebra Morita equivalent to A that has a regular exact Borel subalgebra, but a characterisation of such a Morita representative is not directly obtainable from their work. This paper gives a criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and provides a method to compute all the representatives of A that have a regular exact Borel subalgebra. It is shown that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra (A, Φ, ✂) only depends on the composition factors of the standard and costandard A-modules and on the dimension of the Hom-spaces between standard A-modules. We also characterise the basic quasihereditary algebras that admit a regular exact Borel subalgebra.

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Section: ]) There Is a One-to-one Correspondence Between Regular Dire...mentioning

confidence: 80%

Section: How Bocses Relate To Quasihereditary Algebras and Exact Bore...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

All quasihereditary algebras with a regular exact Borel subalgebra

Conde

2020

Preprint

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“…Bocses are sometimes also called corings. For further reading, we refer to the survey article [Kül17], for general theory of bocses or corings to [BW03], [BSZ09], and [Bur05].…”

Section: Directed Bocsesmentioning

confidence: 99%

Ringel duality as an instance of Koszul duality

Bodzenta

Külshammer

2018

Journal of Algebra

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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

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“…For another interesting approach towards highest weight categories via A ∞categories and bocses we refer to recent work in [24,28].…”

Section: Introductionmentioning

confidence: 99%

In the bocs seat: Quasi-hereditary algebras and representation type

Cited by 10 publications

References 54 publications

All quasihereditary algebras with a regular exact Borel subalgebra

All quasihereditary algebras with a regular exact Borel subalgebra

Ringel duality as an instance of Koszul duality

Highest weight categories and strict polynomial functors. With an appendix by Cosima Aquilino

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