From Levi's Theorem it is known that every finite-dimensional Lie algebra over a field of characteristic zero is decomposed into semidirect sum of its solvable radical and semisimple subalgebra. ...Moreover, semisimple part is the direct sum of simple ideals. In Barnes (preprint) 6 Levi's Theorem is extended to the case of Leibniz algebras. In the present paper we investigate the semisimple Leibniz algebras and we show that the splitting theorem for semisimple Leibniz algebras is not true. Moreover, we consider some special classes of the semisimple Leibniz algebras and we find a condition under which they decompose into direct sum of simple ideals.
In this paper, we study an irreducible decomposition structure of the
D
-module direct image
π
+
(
O
c
n
)
for the finite map
π
:
C
n
→
C
n
/
(
S
n
1
×
⋯
×
S
n
r
)
.
We explicitly construct the ...simple components of
π
+
(
O
C
n
)
by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a
D
-module decomposition of the polynomial ring localized at the discriminant of
π
. Furthermore, we study the action of invariant differential operators on the higher Specht polynomials.
RM ALGEBRAS AND COMMUTATIVE MOONS WALENDZIAK, Andrzej
International Electronic Journal of Algebra,
01/2021, Letnik:
29, Številka:
29
Journal Article
Recenzirano
Odprti dostop
Some generalizations of BCI algebras (the RM, BH, CI, BCH,
BH**, BCH**, and *aRM** algebras) satisfying the identity $(x \rightarrow
1)\rightarrow y = (y \rightarrow 1) \rightarrow x$ are considered. ...The connections of these algebras
and various generalizations of commutative groups (such as, for
example, involutive commutative moons and commutative (weakly)
goops) are described. In particular, it is proved that an RM
algebra verifying this identity is equivalent to an involutive
commutative moon.
We introduce and investigate the strong p-semisimple property for some generalizations of BCI algebras. For BCI algebras, the strong p-semisimple property is equivalent to the p-semisimple property. ...We describe the connections of strongly p-semisimple algebras and various generalizations of groups (such as, for example, involutive moons and goops). Moreover, we present some examples of proper strongly p-semisimple algebras.
Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p ≥ 0, let f : Y → X be a smooth proper morphism and x a geometric point on X. We prove that ...the tensor invariants of bounded length ≤ d of π₁(X, x) acting on the étale cohomology groups H*(Yₓ, 𝔽ℓ) are the reduction modulo-ℓ of those of π₁(X, x) acting on H*(Yₓ, ℤℓ) for ℓ greater than a constant depending only on f : Y → X, d. We apply this result to show that the geometric variant with 𝔽ℓ-coefficients of the Grothendieck-Serre semisimplicity conjecture — namely, that π₁(X, x) acts semisimply on H*(Yₓ, 𝔽ℓ) for l ≫ 0 — is equivalent to the condition that the image of π₁(X, x) acting on H*(Yₓ, ℚℓ) is 'almost maximal' (in a precise sense; what we call 'almost hyperspecial') with respect to the group of ℚℓ-points of its Zariski closure. Ultimately, we prove the geometric variant with 𝔽ℓ-coefficients of the Grothendieck-Serre semisimplicity conjecture.
Abstract
This paper presents some generalizations of BCI algebras (the RM, tRM, *RM, RM**, *RM**, aRM**, *aRM**, BCH**, BZ, pre-BZ and pre-BCI algebras). We investigate the p-semisimple property for ...algebras mentioned above; give some examples and display various conditions equivalent to p-semisimplicity. Finally, we present a model of mereology without antisymmetry (NAM) which could represent a tRM algebra.
In this paper we characterize the maximal modular ideals of an evolution algebra $A$ in order to describe its Jacobson radical, Rad$(A)$. We characterize semisimple evolution algebras (i.e. those ...such that Rad$(A)=\{0\})$ as well as radical ones. We introduce two elemental notions of spectrum of an element $a$ in an evolution algebra $A$, namely the spectrum $\sigma^{A}(a)$ and the $m$-spectrum $\sigma _{m}^{A}(a)$ (they coincide for associative algebras, but in general $\sigma ^{A}(a)\subseteq \sigma_{m}^{A}(a),$ and we show examples where the inclusion is strict). We prove that they are non-empty and describe $\sigma ^{A}(a)$ and $\sigma _{m}^{A}(a)$ in terms of the eigenvalues of a suitable matrix related with the structure constants matrix of $A.$ We say $A$ is $m$-semisimple (respectively spectrally semisimple) if zero is the unique ideal contained into the set of $a$ in $A$ such that $\sigma _{m}^{A}(a)=\{0\}$ (respectively $\sigma^{A}(a)=\{0\}$). In contrast to the associative case (where the notions of semisimplicity, spectrally semisimplicty and $m$-semisimplicity are equivalent) we show examples of $m$-semisimple evolution algebras $A$ that, nevertheless, are radical algebras (i.e Rad$(A)=A$). Also some theorems about automatic continuity of homomorphisms will be considered.
The Kumjian-Pask algebra KP(Λ) is a graded algebra associated to a higher-rank graph Λ and is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left ideals of ...KP(Λ), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(Λ) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian-Pask algebra. As a consequence of this structure theorem, every semisimple Kumjian-Pask algebra can be obtained as a Leavitt path algebra of a directed graph.
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