Minimal Idempotent Research Articles

Decompositions of multiplicative semigroups of m-domain rings and reduced Rickart rings

The main result of this article is that the multiplicative semigroup of an m-domain ring is a strong semilattice of certain subsemigroups, each of which turns out to be a \rcancellative\ monoid, and that this presentation of the semigroup as a strong semilattice of \rcancellative\ semigroups is essentially unique. As a consequence, it is shown that, given an m-domain ring $ \ang{R,+,\cdot} $ with the unary operation $ \dop{} $ mapping every element to its minimal idempotent duplicator (in the sense of N.V.~Subrahmanyam), the algebra $ \ang{R,\cdot,\dop{}} $ is a strong semilattice of \rcancellative\ \dsemigroup s (in the sense of T.~Stokes), also essentially unique. Implications for reduced Rickart rings, which can be seen as a subclass of m-domain rings, are also described.

Module operator virtual diagonals on the Fourier algebra of an inverse semigroup

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over $$\ell ^1(E)$$ . This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449–1474, 1995).

E-Symmetric rings

Let [Formula: see text] be a ring and [Formula: see text] an idempotent of [Formula: see text], [Formula: see text] is called an [Formula: see text]-symmetric ring if [Formula: see text] implies [Formula: see text] for all [Formula: see text]. Obviously, [Formula: see text] is a symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring. In this paper, we show that a ring [Formula: see text] is [Formula: see text]-symmetric if and only if [Formula: see text] is left semicentral and [Formula: see text] is symmetric. As an application, we show that a ring [Formula: see text] is left min-abel if and only if [Formula: see text] is [Formula: see text]-symmetric for each left minimal idempotent [Formula: see text] of [Formula: see text]. We also introduce the definition of strongly [Formula: see text]-symmetric ring and prove that [Formula: see text] is a strongly [Formula: see text]-symmetric ring if and only if [Formula: see text] and [Formula: see text] is a symmetric ring. Finally, we introduce [Formula: see text]-reduced ring and study some properties of it.

Minimal idempotents on solvable groups

In this paper, we begin to develop a theory of character sheaves on an affine algebraic group G defined over an algebraically closed field $$\mathtt {k}$$ of characteristic $$p>0$$ using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let l be a prime different from p. Following Boyarchenko and Drinfeld (Sel Math, 2008. doi: 10.1007/s00029-013-0133-7 , arXiv:0810.0794v1 ), we define the notion of an admissible pair on G and the corresponding idempotent in the $$\overline{\mathbb {Q}}_l$$ -linear triangulated braided monoidal category $$\mathscr {D}_G(G)$$ of conjugation equivariant $$\overline{\mathbb {Q}}_l$$ -complexes (under convolution with compact support) and study their properties. In the spirit of Boyarchenko and Drinfeld (2008), we aim to break up the braided monoidal category $$\mathscr {D}_G(G)$$ into smaller and more manageable pieces corresponding to these idempotents in $$\mathscr {D}_G(G)$$ . Drinfeld has conjectured that the idempotent in $$\mathscr {D}_G(G)$$ obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in $$\mathscr {D}_G(G)$$ can be obtained from some admissible pair on G. In this paper, we prove this conjecture in the case when the neutral connected component $$G^\circ \subset G$$ is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the “Heisenberg case.” Moreover, as we will see in this paper, the study of character sheaves in the Heisenberg case may be considered, in a certain sense, as a twisted version of the theory of character sheaves on reductive groups as developed by Lusztig (Adv Math 56, 57, 59, 61, 1985, 1986).

Modular categories associated with unipotent groups

Let \(G\) be a unipotent algebraic group over an algebraically closed field \(\mathtt{k }\) of characteristic \(p>0\) and let \(l\ne p\) be another prime. Let \(e\) be a minimal idempotent in \(\mathcal{D }_G(G)\), the \(\overline{\mathbb{Q }}_l\)-linear triangulated braided monoidal category of \(G\)-equivariant (for the conjugation action) \(\overline{\mathbb{Q }}_l\)-complexes on \(G\) under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to \(G\) and \(e\) a modular category \(\mathcal{M }_{G,e}\). In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class \(\mathfrak{C }_p^{\pm }\).

Parabolic subgroups and algebraic monoids

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid, G its unit group, P a parabolic subgroup of G, and e ∈ M a minimal idempotent. We show that P = C P ( e ) R u ( G ) and that the assignment P ↦ C P ( e ) defines a one-to-one correspondence between parabolic subgroups of G and of C G ( e ) .

The Q-polynomial idempotents of a distance-regular graph

We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d ⩾ 3 . Let E denote a minimal idempotent of Γ which is not the trivial idempotent E 0 . Let { θ i ∗ } i = 0 d denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E ∘ E is a linear combination of E 0 , E, and at most one other minimal idempotent of Γ; (ii) there exists a complex scalar β such that θ i − 1 ∗ − β θ i ∗ + θ i + 1 ∗ is independent of i for 1 ⩽ i ⩽ d − 1 ; (iii) θ i ∗ ≠ θ 0 ∗ for 1 ⩽ i ⩽ d .

More discrete copies of [formula omitted] in [formula omitted

In a previous paper we established that if q is any minimal idempotent in β N , then for all except possibly one p ∈ c ℓ { 2 n : n ∈ N } ∖ N , q + p + q generates an infinite discrete group. Responding to a question of Wis Comfort, we extend this result in two directions. We show on the one hand that for a minimal idempotent q, there is at most one prime r for which there exists p ∈ c ℓ { r n : n ∈ N } ∖ N such that the group generated by q + p + q is not both infinite and discrete. On the other hand, we show that for any p ∈ β N , if p ∈ c ℓ ( n N ) for infinitely many n ∈ N , then there is some minimal idempotent q such that the group generated by q + p + q is infinite and discrete. We also show that if G is a countable discrete group and if p is a right cancelable element of G ∗ , then there is an idempotent q ∈ G ∗ such that q ⋅ p ⋅ q generates a discrete copy of Z in G ∗ . We do not know whether there exist any minimal idempotent q and any p with p ∈ c ℓ ( n N ) for infinitely many n ∈ N such that the group generated by q + p + q is not discrete. We show that if such a “bad” q exists, then there are many of them.

Distance-regular graphs with light tails

Let Γ be a distance-regular graph with valency k ≥ 3 and diameter d ≥ 2 . It is well known that the Schur product E ∘ F of any two minimal idempotents of Γ is a linear combination of minimal idempotents of Γ . Situations where there is a small number of minimal idempotents in the above linear combination can be very interesting, since they usually imply strong structural properties, see for example Q -polynomial graphs, tight graphs in the sense of Jurišić, Koolen and Terwilliger, and 1- or 2-homogeneous graphs in the sense of Nomura. In the case when E = F , the rank one minimal idempotent E 0 is always present in this linear combination and can be the only one only if E = E 0 or E = E d and Γ is bipartite. We study the case when E ∘ E ∈ span { E 0 , H } ∖ span { E 0 } for some minimal idempotent H of Γ . We call a minimal idempotent E with this property a light tail. Let θ be an eigenvalue of Γ not equal to ± k and with multiplicity m . We show that m − k k ≥ − ( θ + 1 ) 2 a 1 ( a 1 + 1 ) ( ( a 1 + 1 ) θ + k ) 2 + k a 1 b 1 . Let E be the minimal idempotent corresponding to θ . The equality case is equivalent to E being a light tail. Two additional characterizations of the case when E is a light tail are given. One involves a connection between two cosine sequences and the other one a parameterization of the intersection numbers of Γ with a 1 and the cosine sequence corresponding to E . We also study distance partitions of vertices with respect to two vertices and show that the distance-regular graphs with light tails are very close to being 1-homogeneous. In particular, their local graphs are strongly regular.

The Lower Socle and Finite Rank Elementary Operators

We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a k , b k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ↦ a 1 xb 1 + ċ + a n xb n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.

The Galois extensions induced by idempotents in a Galois algebra

Let B be a Galois algebra with Galois group G, Jg = {b ∈ B|bx = g(x)b for all x ∈ B} for each g ∈ G, eg the central idempotent such that BJg = Beg, and for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)( = {g ∈ G | g(eK) = eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also given for G(eK) = N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1 − eG).

Regular algebraic monoids

The purpose of this paper is to provide a proper identification of normal, irreducible, regular algebraic monoids (regular is defined in §2). The results of [3,4] suggest that we should be able to find a classification of these monoids in terms of their unit groups, and related toroidal data. And that is what we accomplish here. Assume that M is a normal, regular, algebraic monoid with unit group G . All our algebraic monoids are defined over an algebraically closed field of arbitrary characteristic. Let e ∈M be a minimal idempotent, and define

Induced permutation automata and coverings of strongly connected automata

Let A = ( S, Σ, N) be a strongly connected automaton and e be a minimal idempotent of characteristic semigroup B( A). The unique (up to isomorphism) subgroup eB( A) e is a very important group in automaton automorphism theory. For example, the automorphism group A( A) of A is a homomorphic image of a subgroup of eB( A) e [7, 8]. If H is a subgroup of A( A), then the automorphism group of the factor automaton A H is also homomorphic to a subgroup of eB( A) e. In this paper we show another property of eB( A) e. First, we introduce the induced permutation automaton whose characteristic semigroup is isomorphic to eB( A) e, and the generalized factor automaton. Using these two automata, we construct a cascade product covering of A, if eB( A) e is not {id} (identity permutation group). This is an example of an effective admissible subset system covering [3], as well as a generalization of the result of Krohn et al. [6]which gave a decomposition of A, if the automorphism group of A is not {id}.

The weakly almost periodic compactification of a direct sum of finite groups

The success of the methods of [ 24 ] and [ 4 ] in investigating the structure of the weakly almost periodic compactification w ℕ of the semigroup (ℕ, +) of positive integers has prompted us to see how successful they would be in another context. We consider wG , where G =[oplus ] i ∈ω G i is the direct sum of a sequence of finite groups with its discrete topology. We discover a large class of weakly almost periodic functions on G , and we use them to prove the existence of a large number of long chains of idempotents in wG . However, the closure of any singly generated subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of βℕ, we find subsemigroups S F of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups S F corresponding to families of disjoint subsets F . Usually S F is a very large semigroup, but for some points p ∈βω, S p can be quite small.

Spectral characterization of the socle in Jordan–Banach algebras

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

Semigroups and automorphism groups of strongly connected automata

If an automatonA = (X, I, M) is strongly connected, $$\bar I$$ is its characteristic semigroup ande is a minimal idempotent of $$\bar I$$ , then the automorphism groupG(A) ofA is a homomorphic image of a subgroup of the group e $$\bar I$$ e (Theorem 3.1) and Theorem 3.3 about the necessary and sufficient conditions ofG(A) to be isomorphic to e $$\bar I$$ e Theorems 3.1 and 3.2 make the result by Fleck (1965) more precise. Our method seems completely different from his method. That is, we use the result on regular permutation groups in Burnside's theory of groups of finite order (Theorem 2.1) as new results on the well-known partial ordering of the set of idempotents of a semigroup (Theorems 2.3, 2.5).

Irreducible Algebras of Operators which Contain a Minimal Idempotent

Irreducible Algebras of Operators which Contain a Minimal Idempotent

Irreducible algebras of operators which contain a minimal idempotent.

We prove that when A is a closed subalgebra of the bounded operators on a reflexive Banach space X, which acts irreducibly on X and contains a minimal idempotent, then every bounded operator with finite dimensional range on X is in A. We use this result to prove that every continuous irreducible representation of a GCR-algebra on a Hilbert space H \mathcal {H} is similar to a ∗ ^ \ast -representation on H \mathcal {H} .

Lattice theory for certain classes of rings

We observe that every Boolean ring is aP1-ring, where each element is its own minimal idempotent duplicator; a Boolean ring with unity is also aP2-ring, where each element is the maximal idempotent annihilator of its complement; and thus,P1-rings andP2-rings are generalisations of a Boolean ring in terms of its lattice structure. Then, it is only natural that these two classes of rings have such close affiliations with the lattice structure of a Boolean ring.