Tame algebras and integral quadratic forms.

*(English)*Zbl 0546.16013
Lecture Notes in Mathematics. 1099. Berlin etc.: Springer-Verlag. xiii, 376 p. DM 51.50 (1984).

This is an elegant introduction to some important parts of the new representation theory of finite dimensional algebras developed during the last 15 years. The author presents several methods of the modern representation theory in a unified and intelligible way. Apart from new versions of known results these notes contain the author’s recent original results concerning interesting classes of tame algebras of finite global dimension.

The main results contained in these notes are presented in terms of the following connection between indecomposable modules and quadratic forms. Let \(A\) be a basic finite dimensional K-algebra of finite global dimension over an algebraically closed field \(K\) having \(n\) nonisomorphic simple left \(A\)-modules. For each finite dimensional left \(A\)-module \(M\), the dimension vector \(\dim M\) of \(M\) is the \(n\)-tuple of multiplicities of fixed nonisomorphic simple \(A\)-modules appearing in the composition series of \(M\). Then \(\dim M\) is an element of the Grothendieck group \(K_0(A)\), identified with \(Z^n\), and we have a bilinear form on \(K_0(A)\) given by \[ <\dim M,\dim N> = \sum_{i\geq 0} (-1)^i\dim_k\mathrm{Ext}^i(M,N). \] The Euler characteristic of \(A\) is the quadratic form \(\chi_A\) associated to \(<-,->\). An element \(x\) of \(K_0(A)\) is called a root of \(\chi_A\) provided \(\chi_A(x)=1\) and, in case \(\chi_A\) is semidefinite, \(x\) is called a radical vector provided \(\chi_A(x)=0\).

Finally we say that the category \(A\)-mod of finite dimensional left \(A\)-modules is controlled by \(\chi_A\) provided the following conditions are satisfied:

(1) for any indecomposable module \(M\) in \(A\)-mod, \(\dim M\) is either a connected positive root or a connected positive radical vector of \(\chi_A\);

(2) for any connected positive root \(x\) of \(\chi_A\) there is precisely one isomorphism class of indecomposable \(A\)-modules \(M\) with \(\dim M=x\);

(3) for any connected positive radical vector \(x\) of \(\chi_A\) there is an infinite family of isomorphism classes of indecomposable modules \(M\) in \(A\)-mod with \(\dim M=x\).

The author has exhibited the structure of the module categories of several classes of algebras \(A\) for which \(A\)-mod is controlled by \(\chi_A\). In particular detailed descriptions of \(A\)-mod for such important, with respect to further study, algebras as tame concealed algebras, domestic tubular extensions of tame concealed algebras and tubular algebras are presented. Interesting properties of some subcategories of \(A\)-mod are also discovered.

Parallel to the main consideration of these notes the author has developed the corresponding theory for the subspace categories of vector space categories, one of the main working tools in the representation theory introduced in Kiev by Nazarova, Roĭter and their students.

The first chapter of the notes is devoted to a self-contained study of integral quadratic forms playing an important role in the further considerations. Moreover the author investigates the representation-finite algebras \(A\) for which \(A\)-mod does not contain oriented cycles (directed algebras) by using methods developed in these notes. A new proof of K. Bongartz’s classification [Comment. Math. Helv. 57, 282–330 (1982; Zbl 0502.16022)] of large sincere directed algebras is also presented.

The main results contained in these notes are presented in terms of the following connection between indecomposable modules and quadratic forms. Let \(A\) be a basic finite dimensional K-algebra of finite global dimension over an algebraically closed field \(K\) having \(n\) nonisomorphic simple left \(A\)-modules. For each finite dimensional left \(A\)-module \(M\), the dimension vector \(\dim M\) of \(M\) is the \(n\)-tuple of multiplicities of fixed nonisomorphic simple \(A\)-modules appearing in the composition series of \(M\). Then \(\dim M\) is an element of the Grothendieck group \(K_0(A)\), identified with \(Z^n\), and we have a bilinear form on \(K_0(A)\) given by \[ <\dim M,\dim N> = \sum_{i\geq 0} (-1)^i\dim_k\mathrm{Ext}^i(M,N). \] The Euler characteristic of \(A\) is the quadratic form \(\chi_A\) associated to \(<-,->\). An element \(x\) of \(K_0(A)\) is called a root of \(\chi_A\) provided \(\chi_A(x)=1\) and, in case \(\chi_A\) is semidefinite, \(x\) is called a radical vector provided \(\chi_A(x)=0\).

Finally we say that the category \(A\)-mod of finite dimensional left \(A\)-modules is controlled by \(\chi_A\) provided the following conditions are satisfied:

(1) for any indecomposable module \(M\) in \(A\)-mod, \(\dim M\) is either a connected positive root or a connected positive radical vector of \(\chi_A\);

(2) for any connected positive root \(x\) of \(\chi_A\) there is precisely one isomorphism class of indecomposable \(A\)-modules \(M\) with \(\dim M=x\);

(3) for any connected positive radical vector \(x\) of \(\chi_A\) there is an infinite family of isomorphism classes of indecomposable modules \(M\) in \(A\)-mod with \(\dim M=x\).

The author has exhibited the structure of the module categories of several classes of algebras \(A\) for which \(A\)-mod is controlled by \(\chi_A\). In particular detailed descriptions of \(A\)-mod for such important, with respect to further study, algebras as tame concealed algebras, domestic tubular extensions of tame concealed algebras and tubular algebras are presented. Interesting properties of some subcategories of \(A\)-mod are also discovered.

Parallel to the main consideration of these notes the author has developed the corresponding theory for the subspace categories of vector space categories, one of the main working tools in the representation theory introduced in Kiev by Nazarova, Roĭter and their students.

The first chapter of the notes is devoted to a self-contained study of integral quadratic forms playing an important role in the further considerations. Moreover the author investigates the representation-finite algebras \(A\) for which \(A\)-mod does not contain oriented cycles (directed algebras) by using methods developed in these notes. A new proof of K. Bongartz’s classification [Comment. Math. Helv. 57, 282–330 (1982; Zbl 0502.16022)] of large sincere directed algebras is also presented.

Reviewer: Andrzej Skowroński (Toruń)

##### MSC:

16Gxx | Representation theory of associative rings and algebras |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16P10 | Finite rings and finite-dimensional associative algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16E20 | Grothendieck groups, \(K\)-theory, etc. |

15A63 | Quadratic and bilinear forms, inner products |

11E04 | Quadratic forms over general fields |