Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBlocks in flat families of finite-dimensional algebras
115
0
0.0
(
0
)
Authors
Ulrich Thiel
Ask ChatGPT about the research
No Arabic abstract
We study the behavior of blocks in flat families of finite-dimensional algebras. In a general setting we construct a finite directed graph encoding a stratification of the base scheme according to the block structures of the fibers. This graph can be explicitly obtained when the central characters of simple modules of the generic fiber are known. We show that the block structure of an arbitrary fiber is completely determined by atomic block structures living on the components of a Weil divisor. As a byproduct, we deduce that the number of blocks of fibers defines a lower semicontinuous function on the base scheme. We furthermore discuss how to obtain information about the simple modules in the blocks by generalizing and establishing several properties of decomposition matrices by Geck and Rouquier.
rate research
Read More
The Jacobian algebra arising from a consistent dimer model is a bimodule $3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric singularity. A perfect matching of a dimer model gives the degree making the Jacobian algebra $mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a $2$-representation infinite algebra which is a generalization of a representation infinite hereditary algebra. In this paper, we show that internal perfect matchings, which correspond to toric exceptional divisors on a crepant resolution of a $3$-dimensional Gorenstein toric singularity, characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Moreover, combining this result with the theorems due to Amiot-Iyama-Reiten, we show that the stable category of graded maximal Cohen-Macaulay modules admits a tilting object for any $3$-dimensional Gorenstein toric isolated singularity. We then show that all internal perfect matchings corresponding to the same toric exceptional divisor are transformed into each other using the mutations of perfect matchings, and this induces derived equivalences of $2$-representation infinite algebras.
For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.
Let $k$ be a field and let $Lambda$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^bullet$ of finitely generated $Lambda$-modules has a well-defined versal deformation ring $R(Lambda,V^bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $Lambda$ and another finite dimensional $k$-algebra $Gamma$ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.
For a Lie algebra ${mathcal L}$ with basis ${x_1,x_2,cdots,x_n}$, its associated characteristic polynomial $Q_{{mathcal L}}(z)$ is the determinant of the linear pencil $z_0I+z_1text{ad} x_1+cdots +z_ntext{ad} x_n.$ This paper shows that $Q_{mathcal L}$ is invariant under the automorphism group $text{Aut}({mathcal L}).$ The zero variety and factorization of $Q_{mathcal L}$ reflect the structure of ${mathcal L}$. In the case ${mathcal L}$ is solvable $Q_{mathcal L}$ is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincar{e} polynomial for solvable Lie algebras. Application is given to $1$-dimensional extensions of nilpotent Lie algebras.
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
