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19
The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra
, 2008
"... We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in char ..."
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We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra Λ, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of Λ in characteristic zero. Criteria for the (semi)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.
COMBINATORIAL DERIVED INVARIANTS FOR GENTLE ALGEBRAS
, 2006
"... Abstract. We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stab ..."
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Abstract. We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable AuslanderReiten quiver of its repetitive algebra. As a byproduct we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.
Degenerations for derived categories
 J. Pure Appl. Algebra
"... Abstract. We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the Riedtma ..."
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Abstract. We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the RiedtmannZwara theorem for module varieties. Applications to tilting complexes are given, in particular that any two term tilting complex is determined by its graded module structure.
Rigidity of tilting complexes and derived equivalence for selfinjective algebras, preprint
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The derived Picard group is a locally algebraic group
 Algebr. Represent. Theory
"... Abstract. Let A be a finite dimensional algebra over an algebraically closed field K. The derived Picard group DPicK(A) is the group of twosided tilting complexes over A modulo isomorphism. We prove that DPicK(A) is a locally algebraic group, and its identity component is Out0 K (A). If B is a deri ..."
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Abstract. Let A be a finite dimensional algebra over an algebraically closed field K. The derived Picard group DPicK(A) is the group of twosided tilting complexes over A modulo isomorphism. We prove that DPicK(A) is a locally algebraic group, and its identity component is Out0 K (A). If B is a derived Morita equivalent algebra then DPicK(A) ∼ = DPicK(B) as locally algebraic groups. Our results extend, and are based on, work of HuisgenZimmermann, Saorín and Rouquier. Let A and B be associative algebras with 1 over a field K. We denote by D b (Mod A) the bounded derived category of left Amodules. Let B ◦ be the opposite algebra, so an A ⊗K B ◦module is a Kcentral ABbimodule. A twosided tilting complex over (A, B) is a complex T ∈ D b (ModA ⊗K B ◦ ) such there exists a complex T ∨ ∈ D b (Mod B ⊗K A ◦ ) and isomorphisms of the derived tensor products T ⊗ L B T ∨ ∼ = A and T ∨ ⊗ L A T ∼ = B. Twosided tilting complexes were introduced by Rickard in [Rd]. When B = A we write A e: = A ⊗K A ◦. The set
DERIVED TAME AND DERIVED WILD ALGEBRAS
, 2003
"... Abstract. We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived wild algebra is derived wild. ..."
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Abstract. We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived wild algebra is derived wild.
DERIVED CATEGORIES AND LIE ALGEBRAS
, 2006
"... Abstract. We define a topological space for the complexes of fixed dimension vector over the derived category of a finite dimensional algebra with finite global dimension. It has a local structure of affine variety under the action of the algebraic group. We consider the constructible functions whic ..."
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Abstract. We define a topological space for the complexes of fixed dimension vector over the derived category of a finite dimensional algebra with finite global dimension. It has a local structure of affine variety under the action of the algebraic group. We consider the constructible functions which reflect the stratifications of the orbit spaces. By using the convolution rule, we obtain the geometric realization of the Lie algebras arising from the derived categories. 1.