Resumen: Let g be a 2n-dimensional unimodular Lie algebra equipped with a Hermitian structure (J; F) such that the complex structure J is abelian and the fundamental form F is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of SU(n ?? k), being 2k the dimension of the center of g. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent. Idioma: Inglés DOI: 10.1007/S00031-015-9352-7 Año: 2016 Publicado en: TRANSFORMATION GROUPS 21, 4 (2016), 903-927 ISSN: 1083-4362 Factor impacto JCR: 0.583 (2016) Categ. JCR: MATHEMATICS rank: 179 / 310 = 0.577 (2016) - Q3 - T2 Factor impacto SCIMAGO: 1.403 - Geometry and Topology (Q1) - Algebra and Number Theory (Q1)