2015/04/15 by Andrew J. Talian, Talian, Andrew J.
Chemistry · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Cyclopropane Reaction Mechanisms #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1504.04059
17 pages
arxiv created 2015/04/15 · openalex publication_date 2015/04/15 · arxiv updated 2015/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If \mathfrakg = \mathfrakg_0 ⊕ \mathfrakg_1 is a Lie superalgebra over an algebraically closed field k of characteristic 0, the notion of an endotrivial module has recently been extended to \mathfrakg-modules by defining M to be endotrivial if Homk(M,M) ≅ kev ⊕ P as \mathfrakg-supermodules. Here, kev denotes the trivial module concentrated in degree 0 and P is a (U(\mathfrakg), U(\mathfrakg_0))-projective supermodule. In the stable module category, these modules form a group under the tensor product. If T(\mathfrakg) denotes the group of endotrivial \mathfrakg-modules, it is interesting and useful to identify this group for a given Lie superalgebra \mathfrakg. In this paper, a classification is given in the case where \mathfrakg = \mathfrakgl(m|n) and it is shown that T(\mathfrakgl(m|n)) ≅ k × ℤ × ℤ2 and is generated by the one parameter family of one dimensional modules kλ where λ∈ k, Ω1(kev), which denotes the first syzygy of kev, and the parity change functor.