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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.
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Let $mathbf{k}$ be field of arbitrary characteristic and let $Lambda$ be a finite dimensional $mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^bullet$ is an object of the bounded derived category $mathcal{D}^b(Lambdatextup{-mod})$ of $Lambda$, then $V^bullet$ has a well-defined versal deformation ring $R(Lambda, V^bullet)$, which is complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$, and which is universal provided that $textup{Hom}_{mathcal{D}^b(Lambdatextup{-mod})}(V^bullet, V^bullet)=mathbf{k}$. Let $mathcal{D}_textup{sg}(Lambdatextup{-mod})$ denote the singularity category of $Lambda$ and assume that $V^bullet$ is a bounded complex whose terms are all finitely generated Gorenstein projective left $Lambda$-modules. In this article we prove that if $textup{Hom}_{mathcal{D}_textup{sg}(Lambdatextup{-mod})}(V^bullet, V^bullet)=mathbf{k}$, then the versal deformation ring $R(Lambda, V^bullet)$ is universal. We also prove that certain singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism class of versal deformation rings of bounded complexes whose terms are finitely generated Gorenstein projective $Lambda$-modules.
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.
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.
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.
We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztigs small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.
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