Nakayama automorphisms of Frobenius algebras

Murray, Will

doi:10.1016/s0021-8693(03)00465-4

Cited by 7 publications

(7 citation statements)

References 6 publications

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“…Self-injective algebras. We need some facts and notation for Frobenius and self-injective algebras, see for instance [Mur03] or [HZ11]. An algebra Λ is self-injective if it is injective as a right Λ-module.…”

Section: Notation 34mentioning

confidence: 99%

Skew group algebras of Jacobian algebras

Giovannini

Pasquali

2019

Journal of Algebra

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For a quiver with potential (Q, W ) with an action of a finite cyclic group G, we study the skew group algebra ΛG of the Jacobian algebra Λ = P(Q, W ). By a result of Reiten and Riedtmann, the quiver Q G of a basic algebra η(ΛG)η Morita equivalent to ΛG is known. Under some assumptions on the action of G, we explicitly construct a potential W G on Q G such that η(ΛG)η ∼ = P(Q G , W G ). The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of G. If Λ is self-injective, then ΛG is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on (Q, W ) behave with respect to our construction.2010 Mathematics Subject Classification. 16G20, 16S35.

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Section: Notation 34mentioning

confidence: 99%

Skew group algebras of Jacobian algebras

Giovannini

Pasquali

2019

Journal of Algebra

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“…[3,13,20]). To be able to formulate this isomorphism property in other monoidal categories C we need a notion of dual object.…”

Section: Frobenius Algebras and Ribbon Categoriesmentioning

confidence: 99%

“…That (4.3) characterizes a symmetric algebra means that ω N is a Nakayama automorphism (see e.g. [14,20]). Any other Nakayama automorphism differs from ω N by an inner automorphism, that is, by a morphism of the form…”

Section: Nakayama Automorphisms and Symmetric Algebrasmentioning

confidence: 99%

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

FUCHS¹

2006

JNMP

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The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms of Frobenius algebras are introduced; they are inner iff the algebra is symmetric. Algebras in monoidal categoriesA (unital, associative) algebra is a triple A = (Ȧ, m, η) consisting of a vecor spaceȦ over some field (or more generally, commutative ring) k, a bilinear map m :Ȧ ×Ȧ →Ȧ and an element e ∈Ȧ such that the associativity and unit propertiesfor all a, b, c ∈Ȧ and m(e, a) = a = m(a, e) for all a ∈Ȧ (1.1)hold. The datum η in the triple A is the linear map from k toȦ that acts asIt is convenient to regard m not as a bilinear map toȦ from the Kronecker productȦ ×Ȧ, but as a linear map toȦ from the tensor productȦ ⊗ kȦ . In terms of the linear maps m and η, the axioms (1.1) of A readIt would actually be more precise to call the structure just described an algebra in the category Vect k of finite-dimensional k-vector spaces. When this formulation is adopted, the requirement of linearity of the maps η and m is already built in automatically, namely by merely demanding that they are allowed maps at all, i.e. that they are morphisms m ∈ Hom(Ȧ ⊗Ȧ,Ȧ) and η ∈ Hom(1,Ȧ) (1.4)

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“…For any finite-dimensional algebra R, the dual R is isomorphic as a left Rmodule to the injective hull of R/radR, so the isomorphism type of R R does not depend on the ground field k. In ( [6]) it is shown that the isomorphism type of R as an (R, R)-bimodule is also independent of k; thus we may speak of R being a Frobenius (respectively, symmetric) algebra without reference to the ground field.…”

Section: Preliminaries and Examplesmentioning

confidence: 99%

Bilinear forms on Frobenius algebras

Murray

2005

Journal of Algebra

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We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius k-algebra R with residue field k. If R is symmetric, then there exists a unique form on R up to homothety iff R is commutative. If R is Frobenius, then we introduce a norm based on the Nakayama automorphism of R. We show that if two forms on R are homothetic, then the norm of the unit separating them is central, and we conjecture the converse. We show that if the dimension of R is even, then the determinant of a form on R, taken ink/k 2 , is an invariant for R.  2005 Elsevier Inc. All rights reserved.

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Nakayama automorphisms of Frobenius algebras

Cited by 7 publications

References 6 publications

Skew group algebras of Jacobian algebras

Skew group algebras of Jacobian algebras

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

Bilinear forms on Frobenius algebras

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