Let R be a left pure semisimple ring such that there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, and let W be the left key R-module; i.e., W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R-modules. We show that if the module W is endofinite, then R is a ring...

In this paper we begin a classification of simple and semisimple totally antiflexible algebras (finite-dimensional) over splitting fields of char. ^2, 3.. For such an algebra A , let P be the largest associative ideal in A and let /V+ be the radical of P. We determine all simple and semisimple totally antiflexible algebras in which N ■ N =0. Defining A to be of type (m, n) if N is nilpotent of ...

Shiquan Ren Februry, 2010 Abstract: I want to write a summary of my study of complex semisimple Lie algebras, as my thesis for a bachelor degree. The classification of complex semisimple Lie algebras is obtained through the classification of simple root systems. Parallel to it, the classification of compact real Lie algebras can be obtained in a similar way. Only after these two classifications...

Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identi...

Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identi...

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds the degrees of minimal generators invariant subring, syzygies modules over subring. These results are analogues for group commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, Symonds.

Let H be a Hopf algebra and A an H-simple right H-comodule algebra. It is shown that under certain hypotheses every (H,A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its f...

In this paper, we consider a generalization of Ebenbauer’s differential equation for non-symmetric matrix diagonalization to a flow on arbitrary complex semisimple Lie algebras. The flow is designed in such a way that the desired diagonalizations are precisely the equilibrium points in a given Cartan subalgebra. We characterize the set of all equilibria and establish a Morse-Bott type property ...

To each real semisimple Jordan algebra, the Tits-Koecher-Kantor theory associates a distinguished parabolic subgroup P = L N of a semisimple Lie group G. The groups P which arise in this manner are precisely those for which N is abelian, and P is conjugate to its opposite P. Each non-open L-orbit O on N∗ admits an L-equivariant measure dμ which is unique up to scalar multiple. By Mackey theory,...

In this lecture I will explain the classification of finite dimensional semisimple Lie algebras over C. Semisimple Lie algebras are defined similarly to semisimple finite dimensional associative algebras but are far more interesting and rich. The classification reduces to that of simple Lie algebras (i.e., Lie algebras with non-zero bracket and no proper ideals). The classification (initially d...