Unital Algebra Research Articles

Graded homotopy classification of Leavitt path algebras over finite primitive graphs

We show that the graded Grothendieck group classifies unital Leavitt path algebras of primitive graphs up to graded homotopy equivalence. To this end, we further develop classification techniques for Leavitt path algebras by means of (graded, bivariant) algebraic K -theory.

Lie theoretic approach to unitary groups of 𝐶*-algebras

Following Robert’s [J. Reine Angew. Math. 756 (2019), pp. 285–319], we study the structure of unitary groups and groups of approximately inner automorphisms of unital C ∗ C^* -algebras, taking advantage of the former being Banach-Lie groups. For a given unital C ∗ C^* -algebra A A , we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by U A U_A , resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by V A V_A , in terms of perfect ideals, i.e. ideals admitting no characters. When the unital algebra is locally AF, we show that there is a one-to-one correspondence between closed normal subgroups of V A V_A and perfect ideals of the algebra, which can be in the separable case conveniently described using Bratteli diagrams; in particular showing that every closed normal subgroup of V A V_A is perfect. We also characterize unital C ∗ C^* -algebras A A such that U A U_A , resp. V A V_A are topologically simple, generalizing the main results of Robert [J. Reine Angew. Math. 756 (2019), pp. 285–319] from \cite{Rob19}. In the other way round, under certain conditions, we characterize simplicity of the algebra in terms of the structure of the unitary group. This in particular applies to reduced group C ∗ C^* -algebras of discrete groups and we show that when A A is a reduced group C ∗ C^* -algebra of a non-amenable countable discrete group, then A A is simple if and only if U A / T U_A/\mathbb {T} is topologically simple.

Description of Local Derivations on Jordan Algebras of Dimension Five

In the present paper we investigate local derivations on finite dimensional Jordan algebras. The Gleason--Kahane--\.{Z}elazko theorem, which is a fundamental contribution in the theory of Banach algebras, asserts that every unital linear functional $F$ on a complex unital Banach algebra $A$, such that $F(a)$ belongs to the spectrum $\sigma(a)$ of $a$ for every $a\in A,$ is multiplicative. In modern terminology this is equivalent to the following condition: every unital linear local homomorphism from a unital complex Banach algebra $A$ into ${\Bbb C}$ is multiplicative. We recall that a linear map $T$ from a Banach algebra $A$ into a~Banach algebra $B$ is said to be a~local homomorphism if, for every $a$ in $A$, there exists a homomorphism $\Phi_a : A\to B$, depending on $a$, such that $T(a)=\Phi_a(a)$. A similar notion was introduced and studied to give a characterization of derivations on operator algebras. Namely, the concept of local derivations was introduced by R.~Kadison and D.~Larson, A.~Sourour independently in 1990. R.~Kadison gave the description of all continuous local derivations from a von Neumann algebra into its dual Banach bemodule. B. Jonson extends the result of R. Kadison by proving that every local derivation from a $C^*$-algebra into its Banach bimodule is a derivation. It is known that, every local derivation on a JB-algebra is a derivation. In particular, every local derivation on a finite dimensional semisimple Jordan algebra is a derivation. In the present paper we investigate derivations and local derivations on five-dimensional nilpotent non-associative Jordan algebras. The description of local derivations of nilpotent Jordan algebras is an open problem. In~the~present paper we give the description of local derivations on five-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of characteristic $\neq 2$, $3$. We also give a~criterion of a~linear operator on Jordan algebras of dimension five to be a local derivation.

The socle of subshift algebras, with applications to subshift conjugacy

We introduce the concept of ‘irrational paths’ for a given subshift and useit to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott–Tomforde–Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.

On the Algebra Generated by Three Commuting Matrices: Combinatorial Cases

Gerstenhaber proved in 1961 that the unital algebra generated by a pair of commuting $d \times d$ matrices over a field has dimension at most $d$. It is an open problem whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for special classes of triples of matrices arising from combinatorial data.

Cofree Com-PreLie algebras

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and the coproduct. We here give examples of cofree Com-PreLie bialgebras, including all the ones such that the preLie product is homogeneous of degree $\geq -1$. We also give a graphical description of free unitary Com-PreLie algebras, explicit their canonical bialgebra structure and exhibit with the help of a rigidity theorem certain cofree quotients, including the Connes-Kreimer Hopf algebra of rooted trees. We finally prove that the dual of these bialgebras are also enveloping algebras of preLie algebras, combinatorially described.

Subalgebras of octonion algebras

For an arbitrary unitary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms.

Higgs branch of 6D (1, 0) SCFTs and little string theories with Dynkin DE-type SUSY enhancement

We detail the Higgs branches of 6D (1, 0) superconformal field theories (SCFTs) and little string theories (LSTs) that exhibit supersymmetry-enhancing Higgs branch renormalization group flows to the 6D (2, 0) SCFTs and LSTs of type DE. Generically, such theories are geometrically engineered in F-theory via a configuration of (−2)-curves, arranged in an (affine) DE-type Dynkin diagram, and supporting special unitary gauge algebras; this describes the effective field theory on the tensor branch of the SCFT. For the Higgsable to D-type (2, 0) SCFTs/LSTs, there generically also exists a type IIA brane description, involving a Neveu-Schwarz orientifold plane, which allows for the derivation of a magnetic quiver for the Higgs branch. These are 3D N=4 unitary-orthosymplectic quivers whose Coulomb branch is isomorphic to the Higgs branch of the 6D theories. From this magnetic quiver, together with an extended quiver subtraction algorithm that we explain, the foliation structure of the Higgs branch as a symplectic singularity is unveiled. For this class of 6D SCFTs, we observe a simple rule, which we refer to as “slice subtraction,” to read off the transverse slice in the foliation from the tensor branch. Based on this slice subtraction observation, we conjecture the transverse slices in the Higgsable to E-type (2, 0) Hasse diagram, where the SCFTs lack any known magnetic quiver for their Higgs branches. Published by the American Physical Society 2024

Integrable couplings stemming from three-dimensional unital algebras

This study aims to explore the connection between integrable couplings and three-dimensional unital algebras. Expansions in these algebras generate integrable couplings, including nonlinear integrable couplings, and their corresponding Lax pairs and hereditary recursion operators possess specific structures, associated with non-semisimple Lie algebras. Applications to the KdV equation are explicitly presented.

Analysis of Complex Entities in Algebra B

We present an exploration of algebra B, a recently published unital and non-associative algebra. Unique complex entities emerged from this algebra, distinct from both quaternion and complex number systems, which we termed treons. We defined the treonic number system and established an isomorphism between this system and the real vector space. Our findings revealed that the treonic representation maintained structural integrity under the defined operations. Based on this foundation, we proceed to determine Euler identity for algebra B within the treonic system. By presenting the fundamental definitions and properties of algebra B, we derived a generalized version of Euler identity applicable within this algebra. This formula revealed the emergence of hyperbolic trigonometric entities, extending the applicability of Euler identity beyond traditional complex numbers. Our results provide a theoretical foundation for a deeper understanding of the properties and behaviors of complex entities within this expanded algebraic framework, thus enabling new theoretical developments and practical applications in the realms of advanced mathematics and theoretical physics.

Left and right Browder elements in Banach algebras

In this paper, we introduce the left (resp. right) Browder elements in a semiprime, complex and unital Banach algebra A. It is established that (i) a∈A is left Browder if and only if a is relatively regular and the left quasinilpotent part of a (or, the left hyperkernel of a) is of finite order; (ii) the spectral mapping theorem holds for the left Browder spectrum; (iii) the set of all left Browder elements is an open multiplicative semigroup; (iv) the left Browder spectrum is stable under commuting Riesz perturbations; (v) Riesz elements are characterized by means of the invariance of the left Browder spectrum under commuting perturbations; (vi) the Kato decomposition for left Browder elements in primitive C* algebras is provided. Similar results hold for the right Browder elements and the right Browder spectrum.

Rationality of holomorphic vertex operator algebras

We prove that if V is a unitary simple holomorphic vertex operator algebra of CFT-type, then V is rational, that is, all N-gradable V-modules are direct sums of copies of V.

Integrable Couplings and Two-Dimensional Unital Algebras

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored.

A categories equivalence of associative bimodules

In this paper we use the classical Wedderburn's Kronecker Factorization Theorem to prove that category of bimodules over $B$ and the category of bimodules over $M_{n}(B)$ are equivalent, where $B$ is some unital associative algebra. In addition to this, we classify the irreducible bimodules over $M_{n}(F).$

Lie superderivations on unital algebras with idempotents

Let U be an associative unital algebra containing a non-trivial idempotent e. We consider U as a superalgebra whose Z 2 -grading is induced by e. This paper aims to describe Lie superderivations of U . In particular, we characterize the general form of Lie superderivations of U and apply it to present the necessary and sufficient conditions for a Lie superderivation on U to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their Z 2 -grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras.

Local and 2-local Lie n-derivations of triangular algebras

Let R be a commutative ring with identity, and A , B be unital algebras over R . Let M be a unital ( A , B ) -bimodule, which is faithful as a left A -module and also as a right B -module. Let T = Tri ( A , M , B ) be an ( n − 1 ) -torsion free triangular algebra. In this note, we investigate the structure of local Lie n-derivations under certain mild conditions and 2-local Lie n-derivations under weaker conditions than that of local Lie n-derivations with n ≥ 3 on T . Under their corresponding conditions, we conclude that they are both standard on T .

A Generalization of the First Tits Construction

Let F be a field of characteristic, not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing the scalar employed in the tripling process outside of the base field. This yields a new family of non-associative unital algebras which carry a cubic map, and maps that can be viewed as generalized adjoint and generalized trace maps. These maps display properties often similar to the ones in the classical setup. In particular, the cubic norm map permits some kind of weak Jordan composition law.

Albert’s twisted field construction using division algebras with a multiplicative norm

We generalize Albert’s twisted field construction, applying it to unital division algebras with a multiplicative norm. We give conditions for the resulting algebras to be division algebras. Four- and eight-dimensional real unital and non-unital division algebras with large derivation algebras are constructed out of Hamilton’s quaternion and Cayley’s octonion algebra.

Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum

Abstract Let 𝒜 {\mathcal{A}} be a complex unital Banach algebra and let R ⊆ 𝒜 {R\subseteq\mathcal{A}} be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, R D {R^{D}} is constructed as an extension of R to axiomatically study the accumulation of σ R ⁢ ( a ) {\sigma_{R}(a)} for any a ∈ 𝒜 {a\in\mathcal{A}} . At last, several illustrative examples on Banach algebra and operator algebra are provided.

Invariant distributions and the transport twistor space of closed surfaces

AbstractWe study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on are determined by their zeroth and first Fourier modes.