# zbMATH — the first resource for mathematics

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.
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.