This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Introduction to lie algebras and representation theory

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.

Generators and representability of functors in commutative and noncommutative geometry

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ
 r
 , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S

 n
 , the Weyl group of ?=??
 n
 . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ??
 n
 , to the algebra of S

 n
 -invariant differential operators with rational coefficients on the space ℂ
 n
 of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ
 n
 , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S

 n
 . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S

 n
 . Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

Introduction Torsion pairs in abelian and triangulated categories Torsion pairs in pretriangulated categories Compactly generated torsion pairs in triangulated categories Hereditary torsion pairs in triangulated categories Torsion pairs in stable categories Triangulated torsion(-free) classes in stable categories Gorenstein categories and (co)torsion pairs Torsion pairs and closed model structures (Co)torsion pairs and generalized Tate-Vogel cohomology Nakayama categories and Cohen-Macaulay cohomology Bibliography Index.

Homological and Homotopical Aspects of Torsion Theories

Noncommutative Projective Schemes

Let A = 0n S,o-^n be a connected graded /r-algebra and let Gr-A denote the category of graded right /4-modules with morphisms being graded homomorphisms of degree 0. If {r,,| n e 1} is a set of graded A:-linear bijections of degree 0 from A to itself satisfying for all I, m, n e Z and all y e Am, z e Ah we define a new graded associative multiplication * on the underlying graded A-vector space 0 n s > o ^ n >' y*z=yrm(z) for all y e Am, z s A,. The graded algebra with the new multiplication * is called a twisted algebra of A. THEOREM. Let A and B be two connected graded algebras generated in degree 1. Then the categories Gx-A and Gx-B are equivalent if and only if A is isomorphic to a twisted algebra of B. If algebras are noetherian, then Gelfand-Kirillov dimension, global dimension, injective dimension, Krull dimension, and uniform dimension are preserved under twisting. Moreover, we prove the following: THEOREM. The following properties are preserved under twisting for connected graded noetherian algebras: (a) Artin-Schelter Gorenstein (or Artin-Schelter regular); (b) Auslander Gorenstein (or Auslander regular) and Cohen-Macaulay. Some of these results are also generalized to certain semigroup-graded algebras.

James J. Zhang

Papers

Noncommutative Projective Schemes

Twisted Graded Algebras and Equivalences of Graded Categories

Rings with Auslander Dualizing Complexes

Skew Calabi–Yau algebras and homological identities

Bialgebra actions, twists, and universal deformation formulas