Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $(Q, I)$ be a bound quiver such that $A=KQ/I$ has infinite global dimension.

I want to ask the following questionss:

(1) Is the Cartan matrix $C_A$ of $A$ invertible in the matrix ring $M_n(Z)$?

(2) what are the relations between the Cartan matrix ( or Coxeter matrix) of a finite dimensional algebra $B$ and the global dimension of $B$? ( Here $B$ may have infinite global dimension)

share|improve this question

1 Answer 1

up vote 1 down vote accepted

(1) No. Example: $$ \begin{array}{ccc} & \alpha & \\ 1 & \rightleftarrows & 2 \\ & \beta & \end{array}$$ with all paths of length 3 = 0. This is a symmetric algebra wigh Loewy structure $$ \begin{array}{ccc} 1 & & 2\\ 2 & \oplus & 1 \\ 1 & & 2 \end{array}$$ and has infinite global dimension. Its Cartan matrix is $$ \left(\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right) $$ whose determinant is not 1, hence not inveritble in $\mathrm{Mat}_2(\mathbb{Z})$.

(2) In general, I don't know. But this is (partly) known for cellular algebras. (And I think BGG algebras, which always is of finite gloabl dimension, always have Cartan matrix with determinant 1, but I am not 100% sure, nor do I have any reference) See C.C.Xi's lecture notes. http://www.math.jussieu.fr/~keller/ictp2006/lecturenotes/xi.pdf The result is: the global dimension of a cellular algebra is finite if and only if the Cartan matrix has determinant 1.

share|improve this answer
    
Thank you very much, Aaron Is there a finite dimensional algebra $A$ satisfying the following conditions: (1) The Cartan matrix of $A$ is not invertible (2) $A$ has finite globle dimension. –  Aimin Xu Nov 28 '12 at 7:31
    
I can't come up with any example yet, so I would say I don't know. For most interesting algebras (to me), i.e. the class of directed algebras, hereditary algebras, quasi-hereditary algebras. They all have finite global dimension with invertible Cartan matrix. (Also explain why it is hard to come up with one such example...) –  Aaron Nov 28 '12 at 14:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.