I need to know: is representation theory all about Groups?
Is it necessary to be a finite group?
Does representation theory exists without Groups?
For example is there sample where representation is about semi-group?
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asked Aug 29, 2013 at 11:45
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2$\begingroup$ Instead of removing structure, you can add more - there is a huge amount of representation theory developed for rings and algebras (both associative and some non-associative). $\endgroup$– mdpAug 29, 2013 at 12:19
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3 Answers
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Yes, there is a matrix representation theory of semigroups. For a survey, see Representations of semigroups by linear transformations.
The term "representation theory" is much more general, not only for groups. Another well-known example is the representation theory of Lie algebras.
answered Aug 29, 2013 at 12:21
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For a more recent reference, see
J. Almeida, S. W. Margolis, B. Steinberg and M. V. Volkov, Representation theory of finite semigroups, semigroup radicals and formal language theory, Trans. Amer. Math. Soc. 361 (2009), 1429-1461.
This paper contains an extensive bibliography on the representation theory of semigroups.
answered Aug 29, 2013 at 13:12
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The notion of an association scheme is a generalization of the notion of a group. Many group-theoretic concepts — including representation theory — have natural generalizations to association schemes. The canonical reference is the book Theory of Association Schemes by Paul-Hermann Zieschang.