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1028:(i.e. the unnecessary double op) makes it seem like there is an error here. Was the intention to define Quiv as the category of pre-sheaves on Q (i.e. objects covariant functors Q^op \to Set)? If that wasn't the case, why not just reverse the arrows on the category Q to make the introduction of the word pre-sheaves seem a little less awful?
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In this article, I see a lot of statements about quivers which presume knowledge about what a quiver is, but I don't see any definition. What is the difference between a quiver and a directed multigraph? If there is none, why is there a separate article about quivers? This article seems to be mainly
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The last line in this section--'Note that Quiv is the category of presheaves on the opposite category Q^op.'--is confusing. Given that pre-sheaves on Q^op would be the category of contravariant set-valued functors on Q^op, which are just covariant set valued functors on Q, this convoluted language
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For infinite Γ the algebra is said to "have enough units" in wisbauer's text. It's pretty similar to the "ring" of finite dimensional matrices (those that act as 0 on all but a finite dimensional space). The identity matrix isn't included, but every finite dimensional identity matrix is, and for
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A lot of it works if K is a artinian semisimple ring, but beyond that most of the classification differs a great deal. If K is self-injective and noetherian, then there is a nice description of injective and projective objects, but there is no longer any nice theory of gabriel roiter on finite
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representation type. If K has finite injective dimension, then work of enochs and herzog gives a reasonable picture of the injective and projective objects for neotherian quivers. In general though (general ring, general quiver) it is a complete mess.
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When the multiplication is defined by concatenation, what is the addition of two paths in the path algebra? Perhaps you can add the explanation in the article. It would help me a lot to understand the definition of a path algebra. Thank you.
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The sum of the vertices. This is hard to see from the wikipedia article since its description of paths seems to suggest they have length at least 1, but the definition really intends to include paths of length
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Multiplying paths together doesn't sound very well defined, unless
Quivers carry a natural structure for multiplying them, which would make them almost identical to categories, except without identity elements.
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about quiver algebras and quiver representations, so shouln't there be two separate articles about those topics? Also, "head" and "tail" are undefined - I assume it means "target" and "source" of an edge. --
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The multiplication structure is perfectly natural -- it's concatenation. Observe that there are no commutativity restrictions, so quivers have quite a lot less structure than categories.
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Actually the correct word here is not path but walk, since we allow the 'path' to visit vertices more than once. See the definitions of walks and paths in graph theory.
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Can someone clarify for me the statement in the introductory paragraph regarding unit elements? If Γ is finite, what is the unit element? --
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The path algebra kQ is a vector space with basis elements: addition is done in exactly the same way as for any other vector space. If
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Well, so in order to avoid further misunderstandings, I'll just replace "composition" by concatenation. -
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Knowledge. If you would like to participate, please visit the project page, where you can join
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lots of arguments this suffices. In PDE type areas, you talk about compact operators and such.
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