I'm having trouble understanding the definition of Verma Module in wikipedia. It later goes to show that it satisfies what appears to be a universal property (which I'm also having trouble understanding - the page is a notational mess). Surely this can be taken as the definition? And is there a clear way of expressing it?
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$\begingroup$ Well, why is this question down-voted ? Is it due to improper question framing ? $\endgroup$– Shanmukha_SrinivasanDec 3, 2012 at 11:33
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$\begingroup$ I'm glad someone asked, I feared it impolite to ask myself :) $\endgroup$– Mozibur UllahDec 3, 2012 at 12:48
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2$\begingroup$ Mozibur, here are some things that would in my opinion improve your question. (1) Link to the wikipedia article. (2) Say where you've looked other than wikipedia: people taking the time to answer a question want to know that the person asking it has already made a serious effort. (3) What exactly is the universal property that the you think Verma modules satisfy, and what exactly do you not understand? (4) Clarify "Surely this can be taken as the definition?": you can't take a univ property as a definition unless you somehow know that something with that property exists. $\endgroup$– Tom LeinsterDec 3, 2012 at 13:15
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1$\begingroup$ PS - It's not impolite to ask why your question is downvoted or closed. Asking good questions takes skill, care and time, and regulars here will often be happy to explain how you can (in their opinion) improve them. $\endgroup$– Tom LeinsterDec 3, 2012 at 13:19
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1$\begingroup$ @Mozibur Ullah: 1) For an elementary textbook definition, you might better start with math.stackexchange.com. But aside from that, you can't expect to study graduate-level mathematics without some library access (paper, online) plus some experience locating material online. Wikipedia by itself is not enough. There are textbook sources for concepts like "Verma module", but also online lecture notes (e.g., by Dennis Gaitsgory at Harvard). 2) The notion of Verma module involves both construction and characterization (existence and uniqueness), both straightforward apart from style. $\endgroup$– Jim HumphreysDec 3, 2012 at 20:02
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1 Answer
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It may be too optimistic to hope for reconciliation of Wikipedia with reality, but, nevertheless, the issue is probably here-to-stay. The universal highest-weight module $V_\lambda$ with given highest weight $\lambda$ is indeed describable be the expected universal mapping proprty, that it has a unique reasonable map to any Lie algebra module generated by a highest weight vector with that weight.
This is version of Frobenius reciprocity...
Since such a module is anticipated to be "induced", it is not surprising that it admits a construction via tensor products and such.
But, yes, sometimes the Wikipedia descriptions are ... too immediate... although one could have imagined that this increases accessibility.
answered Dec 3, 2012 at 3:05