The algebra of upper n x n upper triangular matrices (which is isomorphic to the algebra of lower triangular matrices) is isomorphic to the path algebra of an equi-oriented quiver of type An. This quiver has n(n+1)/2 indecomposable representations (corresponding to the positive roots of a Lie algebra of type An by Gabriel's Theorem). Let me describe these indecomposables. The upper triangular matrices stabilize a flag

(F0,F1,...,Fn) in kn where k is the base field, and Fi is the set of column vectors where only the first i entries are nonzero. For 0

Thanks Peter Breuer, You are right. In fact I used the usual action of this algebra on the space of column vectors to obtain n(n+1)/2 uniserial (non-isomorphic indecomposable) modules (this number is obviousley less that your number!). Initially I thoought that I obtained n! indecomposable modules but it turns out that what I got was n(n+1)l2. So I thought of asking whether the number of indecomposable modules for this algebra is known.

Thanks Harm Derksen

This is exactly the number which is obtained by using the action of this algebra on the space of column vectors. Thanks for your valuable information.

With indecomposable I do mean "not isomorphic to the direct sum of two nonzero modules". Indecomposable modules that do not have any proper submodule are called irreducible.

We can view the 2x2 matrices in the upper left corner of the upper triangular matrices. This is a submodule  (and left-ideal) of the algebra itself. But it is not indecomposable.

In my notation, it is a direct sum of E0,1 and E0,2.