scholarly journals Conjugacy classes of left ideals of a finite dimensional algebra

2013 ◽  
Vol 57 ◽  
pp. 477-496 ◽  
Author(s):  
D. Popa ◽  
G. Sinnamon
Keyword(s):  
Conjugacy Classes ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

Invariants of finite-dimensional algebras recognized by the semigroup of conjugacy classes of left ideals

2017 ◽  
Vol 16 (10) ◽  
pp. 1750182
Author(s):  
Arkadiusz Mȩcel ◽  
Jan Okniński
Keyword(s):  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Ordinary Quiver

We study the semigroup structure on the set [Formula: see text] of conjugacy classes of left ideals of a finite-dimensional algebra [Formula: see text] over an algebraically closed field [Formula: see text], equipped with the natural multiplication inherited from [Formula: see text], and the structure of the contracted semigroup algebra [Formula: see text]. It is shown that [Formula: see text] has a finite chain of ideals with either nilpotent or completely [Formula: see text]-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver [Formula: see text] of [Formula: see text] is proved to be a subquiver of [Formula: see text], if [Formula: see text] is finite. Moreover, in this case, the structure of [Formula: see text] determines, up to isomorphism, the structure of the algebra [Formula: see text] modulo its Jacobson radical. Combining these results we show that if the semigroup [Formula: see text] is finite, then it determines the structure of any (not necessarily basic) triangular algebra [Formula: see text] which admits a normed presentation.

scholarly journals Duality and socle generators for residual intersections

2019 ◽  
Vol 2019 (756) ◽  
pp. 183-226 ◽  
Author(s):  
David Eisenbud ◽  
Bernd Ulrich
Keyword(s):  
Gorenstein Ring ◽  
Jacobian Determinant ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Duality Results ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Residue Theory

AbstractWe prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

Necessary conditions for power commuting in finite-dimensional algebras over a field

2018 ◽  
Vol 28 (5) ◽  
pp. 339-344
Author(s):  
Andrey V. Zyazin ◽  
Sergey Yu. Katyshev
Keyword(s):  
Necessary Conditions ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Finite Dimensional Algebras

Abstract Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.

scholarly journals Residually finite dimensional algebras and polynomial almost identities

2020 ◽  
pp. 2250038
Author(s):  
Michael Larsen ◽  
Aner Shalev
Keyword(s):  
Lie Algebra ◽  
Finite Index ◽  
Open Group ◽  
Finite Codimension ◽  
Finite Dimensional Algebra ◽  
Nilpotent Ideal ◽  
Dimensional Algebra ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

scholarly journals On the finiteness of the derived equivalence classes of some stable endomorphism rings

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1157-1183 ◽  
Author(s):  
Jenny August
Keyword(s):  
Equivalence Class ◽  
Minimal Model ◽  
Equivalence Classes ◽  
Finite Dimensional Algebra ◽  
Derived Equivalence ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Rigid Objects ◽  
Finite Dimensional ◽  
Frobenius Category

Abstract We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $$f :X \rightarrow {\mathrm{Spec}\;}\,R$$ f : X → Spec R , where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $$A_{{\text {con}}}$$ A con known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of $$A_{\text {con}}$$ A con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

Isotopes of simple algebras of arbitrary dimension

2019 ◽  
Vol 13 (06) ◽  
pp. 2050108 ◽  
Author(s):  
Vita Glizburg ◽  
Sergey Pchelintsev
Keyword(s):  
Jordan Algebra ◽  
Matrix Algebra ◽  
Dimensional Space ◽  
Arbitrary Dimension ◽  
Simple Algebra ◽  
Symmetric Bilinear Form ◽  
Finite Dimensional Algebra ◽  
Ground Field ◽  
Dimensional Algebra ◽  
Finite Dimensional

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

Hopf algebra actions and transfer of Frobenius and symmetric properties

2020 ◽  
Vol 126 (1) ◽  
pp. 32-40
Author(s):  
S. Dăscălescu ◽  
C. Năstăsescu ◽  
L. Năstăsescu
Keyword(s):  
Hopf Algebra ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Finite Dimensional Hopf Algebra ◽  
Symmetric Properties

If $H$ is a finite-dimensional Hopf algebra acting on a finite-dimensional algebra $A$, we investigate the transfer of the Frobenius and symmetric properties through the algebra extensions $A^H\subset A\subset A\mathbin{\#} H$.

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