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Address: Liping Li,
School of Mathematics,
University of Minnesota,
504 Vincent Hall, 206 Church St. SE.
Minneapolis, MN 55455
Phone: 612-624-1543(O)
Email: lixxx480@math.umn.edu
Office: 504 Vincent Hall
My son: pic1
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I'm a graduate student at the Department of Mathematics, University of Minnesota. My advisor is Peter Webb. I'll graduate with a Ph.D. in spring 2012.
2. Generalized Koszul Theory
Koszul theory plays an important role in the representation theory of graded algebras. However, there are a lot of structures (algebras, categories, etc) having natural gradings with non-semisimple degree 0 parts. Particular examples include tensor algebras generated by non-semisimple algebras $A_0$ and $(A_0, A_0)$-bimodules $A_1$, and extension algebras of standard modules of standardly stratified algebras. Thus we are motivated to develop a generalized Koszul theory which can be used to study above structures and preserves many classical results such as the Koszul duality. Moreover, we also hope to get a close relation between this generalized theory and the classical theory.
• A generalized Koszul theory and its application , submitted.
Abstract: Let A = ⊕ _{i ≥ 0} A_i be a locally finite algebra, i.e., dim_k A_i < ∞ for all i ≥ 0. In this paper we develop a generalized Koszul theory by assuming that $A_0$ is self-injective. This generalized theory preserves many classical results such as the Koszul duality. Applications of this generalized theory to directed categories and finite EI categories are described.
• A generalized Koszul theory and its relation to the classical theory, preprint.
Abstract: Let A Let $A = \bigoplus _{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ satisfies a certain splitting condition (which is weaker than the assumption that $A_0$ is self-injective). In this paper we generalize many results of the classical theory. Moreover, we define a quotient graded algebra $\bar{A} = \bigoplus _{i \geqslant 0} \bar{A}_i$ and show that $A$ is a generalized Koszul algebra if and only if $\bar{A}$ is a classical Koszul algebra.
3. Stratification Theory
Let A be a finite-dimensional algebra whose indecomposable projective modules (up to isomorphism) are indexed by a preordered set $(\Lambda, \leqslant)$. The algebra A is standardly stratified if there is a set of
indecomposable modules $\Delta_{\lambda}$ satisifying the following conditions: $P_{\lambda}$ has a filtration by $\Delta_{\mu}$ such that all $\mu \succ \lambda$ except one $\lambda$; all composition factors $S_{\mu}$ of
$\Delta_{\lambda}$ satisfy $\mu \preccurlyeq \lambda$. It is quasi-hereditary if furthermore End$_A (\Delta _{\lambda}) \cong k$ for all $\lambda \in \Lambda$.
• Extention algebras of standard modules, accepted by Comm. Algebra.
Abstract: Let A be a basic finite-dimensional k-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $\Gamma = Ext_A(\Delta, \Delta)$ of standard modules, characterize the stratification property of $\Gamma$ for $\leqslant$ and $\leqslant^{op}$, and obtain a sufficient condition for $\Gamma$ to be a generalized Koszul algebra (in a sense which we define).
• Algebras stratified for all linear orders, accepted by Alg. Rep. Theory.
Abstract: In this paper we describe several characterizations of basic finite-dimensional k-algebras A stratified for all linear orders, and classify their graded algebras as tensor algebras satisfying some extra
property. We also discuss whether for a given linear order $\preccurlyeq$, $\mathcal{F} (_{\preccurlyeq} \Delta)$, the category of A-modules with $_{\preccurlyeq} \Delta$-filtrations, is closed under cokernels of monomorphisms, and classify quasi-hereditary algebras satisfying this property.
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