Field-dependent homological behavior of finite dimensional algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

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Levi and Malcev Theorems for Finite-Dimensional Algebras from the Variety Defined by the Identities x2 = J( x,y, zu) =0

Algebra Colloquium ◽

10.1142/s1005386721000080 ◽

2021 ◽

Vol 28 (01) ◽

pp. 87-90

Author(s):

Óscar Guajardo Garza ◽

Marina Rasskazova ◽

Liudmila Sabinina

Keyword(s):

Lie Algebras ◽

Present Article ◽

Variety Of Algebras ◽

Finite Dimensional ◽

Finite Dimensional Algebras

We study the variety of binary Lie algebras defined by the identities [Formula: see text], where [Formula: see text] denotes the Jacobian of [Formula: see text], [Formula: see text], [Formula: see text]. Building on previous work by Carrillo, Rasskazova, Sabinina and Grishkov, in the present article it is shown that the Levi and Malcev theorems hold for this variety of algebras.

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Piecewise Hereditary Triangular Matrix Algebras

Algebra Colloquium ◽

10.1142/s1005386721000134 ◽

2021 ◽

Vol 28 (01) ◽

pp. 143-154

Author(s):

Yiyu Li ◽

Ming Lu

Keyword(s):

Positive Integer ◽

Triangular Matrix ◽

Derived Categories ◽

Matrix Algebras ◽

Tensor Algebras ◽

Finite Dimensional ◽

Abelian Categories ◽

Upper Triangular Matrix ◽

Orbit Categories ◽

Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

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Homological Aspects of Semigroup Gradings on Rings and Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1999-022-5 ◽

1999 ◽

Vol 51 (3) ◽

pp. 488-505 ◽

Cited By ~ 3

Author(s):

W. D. Burgess ◽

Manuel Saorín

Keyword(s):

Euler Characteristic ◽

Division Ring ◽

Artinian Ring ◽

Global Dimension ◽

Monomial Algebra ◽

Finite Dimensional ◽

Cartan Matrices ◽

Torsion Free ◽

Rings And Algebras ◽

Finite Dimensional Algebras

AbstractThis article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup Σ. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the Σ-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert Σ-series in the associated path incidence ring.The rationality of the Σ-Euler characteristic, the Hilbert Σ-series and the Poincaré-Betti Σ-series is studied when Σ is torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

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