Based on the definitions of fuzzy associative algebras and fuzzy ideals, it is proven that the intersections of fuzzy subalgebras are fuzzy subalgebras, and the intersections of fuzzy ideals are ...fuzzy ideals. Moreover, we prove that the kernels of fuzzy homomorphisms are fuzzy ideals. Using fuzzy ideals, the quotient structures of fuzzy associative algebras are constructed, their corresponding properties are discussed, and their homomorphism theorems are proven.
Let
A
be an
n
-dimensional algebra over a field
k
and
a
(
A
) its quantum symmetry semigroup. We prove that the automorphisms group
Aut
Alg
(
A
)
of
A
is isomorphic to the group
U
(
G
(
a
(
A
)
o
)
)
...of all invertible group-like elements of the finite dual
a
(
A
)
o
. For a group
G
, all
G
-gradings on
A
are explicitly described and classified: the set of isomorphisms classes of all
G
-gradings on
A
is in bijection with the quotient set
Hom
BiAlg
(
a
(
A
)
,
k
G
)
/
≈
of all bialgebra maps
a
(
A
)
→
k
G
, via the equivalence relation implemented by the conjugation with an invertible group-like element of
a
(
A
)
o
.
In order to obtain the classification of 4-dimensional associative algebras with principal generators over complex number C under isostructuralism, by using the ring theory and distribution of roots ...of the equations satisfied by the principal generators: one quadruple root, four different roots, one triple root and one simple root, two different double roots, one double root and two different simple roots, the above equations are translated and stretched, then the more simple equations are obtained. By using linear algebras, quotient algebras and maple to make a large number of operations, it is concluded that each isomorphism type has nothing to do with parameters. Finally by comparing each isomorphism type, the complete classification is obtained, which deepens the realization of structure of asso ciative algebra, and provides reference for relative study.
On free subalgebras of varieties Fehlberg Júnior, Renato; Sánchez, Javier
Proceedings of the Edinburgh Mathematical Society,
02/2022, Volume:
65, Issue:
1
Journal Article
Peer reviewed
Open access
We show that some results of L. Makar-Limanov, P. Malcolmson and Z. Reichstein on the existence of free-associative algebras are valid in the more general context of varieties of algebras.
We introduce a non-symmetric operad , whose dimension in degree n is given by the Catalan number c
n−1
. It arises naturally in the study of coalgebra structures defined on compatible associative ...algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a -algebra. The data (As,As
2
, ) is a good triple of operads, in J.-L. Loday's sense. Our construction induces another triple of operads (As,As
2
,As), where As
2
is the operad of matching dialgebras. Motivated by A. Goncharov's Hopf algebra of paths P(S), we introduce the notion of bi-matching dialgebras and show that the Hopf algebra P(S) is a bi-matching dialgebras.
In this paper, Hom-Jordan and Hom-alternative bimodules are introduced. It is shown that Jordan and alternative bimodules are twisted via endomorphisms into Hom-Jordan and Hom-alternative bimodules ...respectively. Some relations between Hom-associative bimodules, Hom-Jordan and Hom-alternative bimodules are given.
In this paper, Hom-Jordan and Hom-alternative bimodules are introduced. It is shown that Jordan and alternative bimodules are twisted via endomorphisms into Hom-Jordan and Hom-alternative bimodules respectively. Some relations between Hom-associative bimodules, Hom-Jordan and Hom-alternative bimodules are given.
We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several ...examples.
We propose a new approach to study the relation between the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B=C⋉E. This new approach consists of using the ...induction functor −⊗CB as well as the coinduction functor D(B⊗CD−). We show that DE is a partial tilting and a τ-rigid C-module and that the induced module DE⊗CB is a partial tilting and a τ-rigid B-module. Furthermore, if C=EndAT for a tilting module T over a hereditary algebra A, we compare the induction and coinduction functors to the Buan–Marsh–Reiten functor HomCA(T,−) from the cluster-category of A to the module category of B. We also study the question as to which B-modules are actually induced or coinduced from a module over a tilted algebra.
PDF
