(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra?

For example, consider this Lie algebra:

$$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad [Y^i,Y^j] = 0 $$ Here, for example, consider 3 generators $X_1,X_2,X_3$ generate a compact semi-simple SU(2) Lie algebra with $f_{ij}{}^k$ given by $f_{12}{}^3=1$ and $f_{23}{}^1=-1$ as $i,j,k$ are cyclic. And another 3 generators $Y^1,Y^2,Y^3$ are Abelian extension of $X_1,X_2,X_3$. (Some people would use the words semi-direct product for the groups $g(X) \ltimes g(Y)^*$.)

(2) What is the Cartan matrix for this Lie algebra above? (for this whole non-semisimple Lie algebra $g(X) \ltimes g(Y)^*$.)

Thank you for any comments and concerns! Please reply whatever thoughts you have.

  • $\begingroup$ Cartan matrix in what sense? The construction of the Cartan matrix of a semisimple Lie algebra depends quite muchly on the semisimpleness of the algebra (to get an abelian Cartan subalgebra, and so on...) $\endgroup$ – Mariano Suárez-Álvarez Jan 10 '14 at 4:48
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    $\begingroup$ @ Mariano: regarding not semisimple: "In modular representation theory, and more generally in the theory of representations of finite-dimensional associative algebras A that are not semisimple, a Cartan matrix is defined by considering a (finite) set of principal indecomposable modules and writing composition series for them in terms of irreducible modules, yielding a matrix of integers counting the number of occurrences of an irreducible module." from Wiki. $\endgroup$ – wonderich Jan 10 '14 at 20:16
  • $\begingroup$ @Idear: are you sure that this is the same matrix? At first sight it seems to be quite a different object just accidentally having a same name (I'm sure Cartan has looked at more than one matrix during his life...). Also, I think you are doing neither modular representation theory nor working with associative algebras, right? $\endgroup$ – Marek Jan 10 '14 at 21:05
  • $\begingroup$ @ Marek, not sure whether there are the same matrix, but it seems to be a generalized version of Cartan matrix. Please feel free to correct me, I am willing to hear. Thanks. $\endgroup$ – wonderich Jan 10 '14 at 21:20
  • $\begingroup$ @Idear, that matrix is a COMPLETELY different thing from the Cartan matrix of a semisimple algebra. $\endgroup$ – Mariano Suárez-Álvarez Jan 14 '14 at 6:28

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