Abstract
We define associative algebras over fields. Roughly speaking, an algebra is a ring which also is a vector space such that scalars commute with everything. One example are the n × n-matrices over some field, with the usual matrix addition and multiplication. We introduce many examples of algebras, these include three types of algebras which will play a special role in the following, namely division algebras, group algebras, and path algebras of quivers. We start to develop the general theory of algebras by studying subalgebras, factor algebras and algebra homomorphisms.
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K. Erdmann, M.J. Wildon, Introduction to Lie Algebras. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2006. x+251 pp.
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Erdmann, K., Holm, T. (2018). Algebras. In: Algebras and Representation Theory. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-91998-0_1
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DOI: https://doi.org/10.1007/978-3-319-91998-0_1
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Online ISBN: 978-3-319-91998-0
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