Titlebook: Algebras and Representation Theory; Karin Erdmann,Thorsten Holm Textbook 2018 Springer Nature Switzerland AG 2018 MSC (2010) 16-XX, 16G10,

约会

,The Structure of Semisimple Algebras: The Artin–Wedderburn Theorem, matrix algebras are semisimple algebras. When the coefficient field is algebraically closed the Artin-Wedderburn theorem shows that every semisimple algebra is isomorphic to such an algebra. In general, a semisimple algebra is isomorphic to a direct product of matrix algebras but where matrix blocks over division algebras may occur.

decode

,Semisimple Group Algebras and Maschke’s Theorem,over a field is semisimple if and only the characteristic of the coefficient field does not divide the order of the group. We prove this theorem, and we discuss connections between the Artin-Wedderburn decomposition of a semisimple group algebra and properties of the group.

ARM

https://doi.org/10.1057/9781137343215We discuss tools to find and to compare simple modules of a fixed algebra. Furthermore, we determine the simple modules for finite-dimensional quotients of polynomial algebras in one variable, and also for finite-dimensional path algebras of quivers. Finally, we prove Schur’s lemma and discuss some applications.

TAP

engender

https://doi.org/10.1057/9781137277121sentially unique; this is known as the Krull-Schmidt Theorem, we give a proof. One may think of simple modules as analogs of ‘elementary particles’, and then indecomposable modules could be viewed as analogs of ‘molecules’.

胆大

Georgios Anagnostopoulos,Gerasimos Santasr some classes of algebras. In particular, we show that the group algebra of a finite group has finite representation type if and only if the Sylow .-subgroups of the group are cyclic, here . is the characteristic of the coefficient field.

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COKE

,Simple Modules and the Jordan–Hölder Theorem,We discuss tools to find and to compare simple modules of a fixed algebra. Furthermore, we determine the simple modules for finite-dimensional quotients of polynomial algebras in one variable, and also for finite-dimensional path algebras of quivers. Finally, we prove Schur’s lemma and discuss some applications.

HEDGE

Semisimple Modules and Semisimple Algebras,as an exception, we deal with arbitrary direct sums of modules. We view this as an opportunity to prove a result which does not need finiteness assumptions. The only new tool necessary is Zorn’s lemma.

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