Geodesic
Two-term partial tilting complexes over Brauer tree algebras
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 24, Tome 413 (2013), pp. 5-25

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In this paper, we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings. 
@article{ZNSL_2013_413_a0,
     author = {M. A. Antipov and A. O. Zvonareva},
     title = {Two-term partial tilting complexes over {Brauer} tree algebras},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--25},
     publisher = {mathdoc},
     volume = {413},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_413_a0/}
}
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M. A. Antipov; A. O. Zvonareva. Two-term partial tilting complexes over Brauer tree algebras. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 24, Tome 413 (2013), pp. 5-25. http://geodesic.mathdoc.fr/item/ZNSL_2013_413_a0/