Quadratic algebras.

*(English)*Zbl 1145.16009
University Lecture Series 37. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3834-2/pbk). xii, 159 p. (2005).

An associative algebra \(A\) over a field \(k\) is quadratic if there exists a finite subset in \(A\) with quadratic homogeneous defining relations. The purpose of the book is to present some new recent results of quadratic algebras.

Each quadratic algebra \(A\) is isomorphic to a factoralgebra \(T(V)/I\) where \(T(V)\) is the tensor algebra of a finite dimensional algebra over a finite dimensional vector space \(V\) and \(I\) is an ideal in \(T(V)\) generated by \(I\cap(V\otimes V)\). If this presentation is fixed then the dual quadratic algebra \(A^!=T(V^*)/I^\perp\), where \(V^*\) is the dual space of \(V\) and \(I^\perp\) as an ideal is generated by the set of all \(l\in(V^*\otimes V^*)=(V\otimes V)^*\) such that \(l(I\cup(V\otimes V))=0\). A quadratic algebra \(A\) is Koszul if \(\text{Ext}^*(k,k)=A^!\).

Section 2 deals with Hilbert series of Koszul algebras, distributive subspaces in \(T(V)\), behavior of Koszul algebras under homomorphisms, connections with algebraic geometry. It is shown that the category of Koszul algebras in a \(k\)-linear monoidal Abelian category is equivalent to a category of exact monoidal functors from a special category to \(C\).

Chapter 3 deals with Segre products \(A\circ B\) and Veronese powers \(A^{(d)}\). It is mentioned that classes of quadratic (Koszul) algebras are closed under Serge products and Veronese powers. An adjoint functor to Serge product \(A\circ B\) is equal to \(\text{cohom}(A,B)\). Various properties of \(\text{cohom}\) are established in Chapter 3. It is mentioned that there exists a Koszul algebra \(A\) and a quadratic non-Koszul algebra \(A_1\) such that the Hilbert series of the pairs \(A,A_1\) and of \(A^!,A^!_1\) coincide.

Chapter 4 is devoted to PBW-bases in quadratic algebras. It is shown that the class of these algebras consists of Koszul algebras and it is closed under dual algebras, Serge products and Veronese powers. Some results related to calculations of Hilbert series of an algebra with PBW-bases can be found in Chapter 4.

Section 5 deals with filtered algebras \(A\) generated by \(A_1\) with defining relations of the form \[ \sum_{i,j} \alpha_{ij} x_ix_j+\sum_i\beta_ix_i=0,\quad\alpha_{i,j},\beta_i\in k. \] PBW-bases, dual algebras, bar-complexes are extended to this non-graded case.

In Section 6 the authors consider deformation of Koszul algebras, generic quadratic algebras. Some additional analytic properties of Hilbert series are established in Section 8.

In the introduction the authors mention Chapter 8 and 4 which are not included into the text.

Each quadratic algebra \(A\) is isomorphic to a factoralgebra \(T(V)/I\) where \(T(V)\) is the tensor algebra of a finite dimensional algebra over a finite dimensional vector space \(V\) and \(I\) is an ideal in \(T(V)\) generated by \(I\cap(V\otimes V)\). If this presentation is fixed then the dual quadratic algebra \(A^!=T(V^*)/I^\perp\), where \(V^*\) is the dual space of \(V\) and \(I^\perp\) as an ideal is generated by the set of all \(l\in(V^*\otimes V^*)=(V\otimes V)^*\) such that \(l(I\cup(V\otimes V))=0\). A quadratic algebra \(A\) is Koszul if \(\text{Ext}^*(k,k)=A^!\).

Section 2 deals with Hilbert series of Koszul algebras, distributive subspaces in \(T(V)\), behavior of Koszul algebras under homomorphisms, connections with algebraic geometry. It is shown that the category of Koszul algebras in a \(k\)-linear monoidal Abelian category is equivalent to a category of exact monoidal functors from a special category to \(C\).

Chapter 3 deals with Segre products \(A\circ B\) and Veronese powers \(A^{(d)}\). It is mentioned that classes of quadratic (Koszul) algebras are closed under Serge products and Veronese powers. An adjoint functor to Serge product \(A\circ B\) is equal to \(\text{cohom}(A,B)\). Various properties of \(\text{cohom}\) are established in Chapter 3. It is mentioned that there exists a Koszul algebra \(A\) and a quadratic non-Koszul algebra \(A_1\) such that the Hilbert series of the pairs \(A,A_1\) and of \(A^!,A^!_1\) coincide.

Chapter 4 is devoted to PBW-bases in quadratic algebras. It is shown that the class of these algebras consists of Koszul algebras and it is closed under dual algebras, Serge products and Veronese powers. Some results related to calculations of Hilbert series of an algebra with PBW-bases can be found in Chapter 4.

Section 5 deals with filtered algebras \(A\) generated by \(A_1\) with defining relations of the form \[ \sum_{i,j} \alpha_{ij} x_ix_j+\sum_i\beta_ix_i=0,\quad\alpha_{i,j},\beta_i\in k. \] PBW-bases, dual algebras, bar-complexes are extended to this non-graded case.

In Section 6 the authors consider deformation of Koszul algebras, generic quadratic algebras. Some additional analytic properties of Hilbert series are established in Section 8.

In the introduction the authors mention Chapter 8 and 4 which are not included into the text.

Reviewer: Vyacheslav A. Artamonov (Moskva)

##### MSC:

16S37 | Quadratic and Koszul algebras |

16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |

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

16W50 | Graded rings and modules (associative rings and algebras) |

16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |