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?



Are Verma modules universally characterised?
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.


