Algebra Homomorphism Research Articles

Idempotents of large norm and homomorphisms of Fourier algebras

We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms in the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. We prove that the existence of idempotents of arbitrarily large norm in B(G) implies the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for every locally compact group H. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for some amenable locally compact group H implies the existence of idempotents of arbitrarily large norm in B(G).

ALGEBRAIC STRUCTURES BY CREATING MIND MAPS WITH STUDENTS GIFTED IN MATHEMATICS

Working with mathematically gifted students is the subject of many studies. In the literature, one can find various examples of the positive impact of the use of mind maps on learning by understanding and connecting concepts in appropriate schemes, but the impact of creating mind maps on the achievements of students gifted in mathematics has not been sufficiently researched. Having that in mind and that algebraic structures represent a teaching topic in which it is necessary for students to have adequate theoretical knowledge about these structures and relations between them, this method was implemented in order for students to connect the proper concepts in a scheme. For that purpose, (two) mathematics classes of sys- tematization are conducted for the teaching topic on Algebraic structures in order for students to create two mind maps each (one for algebraic structures with one and with two binary operations and another for homomorphism). The effects of this approach to the systematization classes were examined by analyzing the students’ success achieved in two fifteen-minute tests (before and after the systematization classes) where they had to mark the correct statements (precisely formulated algebraic structures and homomorphisms). The results obtained by statistical analysis indicate that the students achieved statistically significantly better results in the test held after the systematization classes. In other words, the creation of mind maps by students gifted in mathematics had a positive effect on systematization of knowledge about Algebraic structures and on students’ achievement in mathematics (specifically Linear Algebra and Analytical Geometry). This result implies that teachers who work with students gifted in mathematics should seriously consider organizing mathematics classes where students will systematize and deepen their theoretical knowledge by creating mind maps.

A new realisation of the i-quantum group U(n)

We follow the approach developed in [3] and modified in [7] to investigate a new realisation for the i -quantum groups U ȷ ( n ) , building on the multiplication formulas discovered in [2, Lem. 3.2] . This allows us to present U ȷ ( n ) via a basis and multiplication formulas by generators. We also establish a surjective algebra homomorphism from a Lusztig type form U Z ȷ ( n ) of U ȷ ( n ) to the integral q -Schur algebras S Z ȷ ( n , r ) of type B . Thus, base change to a field k allows us to relate representations of the i -quantum hyperalgebras U k ȷ ( n ) to representations of finite orthogonal groups of odd degree in non-defining characteristics. This generalises part of Dipper–James' type A theory to the type B case.

Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain $C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. In addition, we discuss the theory of holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results.

A look into homomorphisms between uniform algebras over a Hilbert space

We study the vector-valued spectrum $\mathcal {M}_{u,\infty }(B_{\ell _2},B_{\ell _2})$, which is the set of non-zero algebra homomorphisms from $\mathcal {A}_u(B_{\ell _2})$ (the algebra of uniformly continuous holomorphic functions on $B_{\ell _2}$) to

Linking the special orthogonal algebra [formula omitted] and the tetrahedron algebra ⊠

In 2007, B. Hartwig and Terwilliger found a presentation for the three-point sl2 loop algebra in terms of generators and relations. To obtain this presentation, they defined a Lie algebra ⊠ by generators and relations and established an isomorphism from ⊠ to three-point sl2 loop algebra. Essentially, ⊠ has six generators which can be naturally identified with the six edges of the tetrahedron. In fact, each face of the tetrahedron has three surrounding edges which generate a subalgebra of ⊠ that is isomorphic to sl2. It is interesting to know whether a direct sum of finitely many copies of sl2 (e.g., special orthogonal algebra so4) captures the bracket relations of the generators of ⊠. Here, we show that there exists a Lie algebra homomorphism ϕ:⊠→so4 which can be extended to a homomorphism ϕ:⊠→L where L is a direct sum of finitely many copies of sl2. We construct a finite-dimensional so4-module which is viewed as a ⊠-module via the homomorphism ϕ. We show how this so4-module is related to Krawtchouk polynomials. This paper is inspired by and is an extension of the work of Nomura and Terwilliger (2012) [19].

The SHAI property for the operators on Lp

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for Lp(0,1) for 1<p<∞, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1<p<∞.

Noncommutative differential calculus structure on secondary Hochschild (co)homology

Let B be a commutative algebra and A be a B-algebra (determined by an algebra homomorphism ). M. D. Staic introduced a Hochschild like cohomology called secondary Hochschild cohomology, to describe the non-trivial B-algebra deformations of A. J. Laubacher et al later obtained a natural construction of a new chain complex in the process of introducing the secondary cyclic (co)homology. In this paper, we establish a connection between the two (co)homology theories for B-algebra A. We show that the pair forms a noncommutative differential calculus, where denotes the homology of the complex

An analog of the Feigin-Frenkel homomorphism for double loop algebras

We prove the existence of a homomorphism of vertex algebras, from the vacuum Verma module over the loop algebra of an untwisted affine algebra, whose construction is analogous to that of the Feigin-Frenkel homomorphism from the vacuum Verma module at critical level over an affine algebra.

The complexity of homomorphism factorization

We investigate the computational complexity of the problem of deciding if an algebra homomorphism can be factored through an intermediate algebra. Specifically, we fix an algebraic language, \(\mathcal L\), and take as input an algebra homomorphism \(f:X\rightarrow Z\) between two finite \(\mathcal L\)-algebras X and Z, along with an intermediate finite \(\mathcal L\)-algebra Y. The decision problem asks whether there are homomorphisms \(g:X\rightarrow Y\) and \(h:Y\rightarrow Z\) such that \(f=hg\). We show that this problem is NP-complete for most languages.

Representations of power-associative train algebras of rank 4

This paper is devoted to the study of four-dimensional commutative power-associative non-Jordan train algebras. The existence of a unique orthogonal idempotent element in such kind of algebras enables us to define a non-zero algebra homomorphism over their ground field, which gives rise to train algebras of rank 4. We describe the irreducible representations over an algebraically closed field of characteristic prime to 30 by means of multilinear identities which define it. Furthermore, we provide sufficient conditions to guarantee whenever a representation is decomposable.

The Formal Shift Operator on the Yangian Double

Abstract Let ${\mathfrak{g}}$ be a symmetrizable Kac–Moody algebra with associated Yangian $Y_\hbar{\mathfrak{g}}$ and Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in{\mathbb{C}}$, there is an automorphism $\tau _c$ of $Y_\hbar{\mathfrak{g}}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $\tau _z$ from $Y_\hbar{\mathfrak{g}}$ to the polynomial algebra $Y_\hbar{\mathfrak{g}}[z]$. We prove that $\tau _z$ uniquely extends to an algebra homomorphism $\Phi _z$ from the Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$ into the $\hbar $-adic closure of the algebra of Laurent series in $z^{-1}$ with coefficients in the Yangian $Y_\hbar{\mathfrak{g}}$. This induces, via evaluation at any point $c\in{\mathbb{C}}^\times $, a homomorphism from $\textrm{D}Y_\hbar{\mathfrak{g}}$ into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of $\textrm{D}Y_\hbar{\mathfrak{g}}$ and $Y_\hbar{\mathfrak{g}}$ and, as a corollary, we find that the Yangian $Y_\hbar{\mathfrak{g}}$ can be realized as a degeneration of the Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. Using these results, we obtain a Poincaré–Birkhoff–Witt theorem for $\textrm{D}Y_\hbar{\mathfrak{g}}$ applicable when ${\mathfrak{g}}$ is of finite type or of simply laced affine type.

The Roger–Yang skein algebra and the decorated Teichmüller space

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper, we consider surfaces with punctures which are not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang’s Poisson algebra homomorphism is injective, and the skein algebra has no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.

Mappings between the lattices of saturated submodules with respect to a prime ideal

Let $\mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an $R$-module $M$ with respect to a prime ideal $p$ of a commutative ring $R$. We examine the properties of the mappings $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$, in particular considering when these mappings are lattice homomorphisms. It is proved that if $M$ is a semisimple module or a projective module, then $\eta$ is a lattice homomorphism. Also, if $M$ is a faithful multiplication $R$-module, then $\eta$ is a lattice epimorphism. In particular, if $M$ is a finitely generated faithful multiplication $R$-module, then $\eta$ is a lattice isomorphism and its inverse is $\theta$. It is shown that if $M$ is a distributive module over a semisimple ring $R$, then the lattice $\mathfrak{S}_p(_RM)$ forms a Boolean algebra and $\eta$ is a Boolean algebra homomorphism.

Adding an identity to a Banach lattice algebra: a look at a Wickstead’s counter-example from a norm-free point of view

Recently, Wickstead investigated the long-standing problem of adding an identity to a non-unital Banach lattice algebra. In this regard, he proved that, in the category whose objects are unital Banach lattice algebras and morphisms are identity preserving algebra and lattice homomorphisms, there is no reflection for $$c_{0}$$ in which $$c_{0}$$ can be embedded as an algebra and order ideal. The main purpose of this note is to describe an alternative category in which a suitable reflection of $$c_{0}$$ can be located, producing a satisfactory lattice unitization of $$c_{0}$$ with $$c_{0}$$ as an algebra and order ideal.

The evolution operator of evolution algebras

This paper deals with the evolution operator of evolution algebras. The main goal is to obtain novel results of this operator, both in the case in which it is a derivation and when it is a homomorphism of algebras. These new results constitute a criterion for classifying evolution algebras satisfying one of these conditions. An algorithm to obtain all these degenerate evolution algebras starting from those of smaller dimensions is also constructed.

Surjective Homomorphisms from Algebras of Operators on Long Sequence Spaces are Automatically Injective

Abstract We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on X, to $\mathscr{B}(Y)$, where X is one of the following long sequence spaces: c0(λ), $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p \lt \infty$) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.

Noncommutative differential calculus on (co)homology of hom-associative algebras

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. It was previously shown by the author that the Hochschild cohomology of a hom-associative algebra A carries a Gerstenhaber structure. In this short paper, we show that this Gerstenhaber structure together with certain operations on the Hochschild homology of A makes a noncommutative differential calculus. As an application, we obtain a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of a regular unital symmetric hom-associative algebra.

Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type (C1∨,C1)

Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative F -algebra defined by generators and relations. The generators are A , B , C , D and the relations state that [ A , B ] = [ B , C ] = [ C , A ] = 2 D and each of [ A , D ] + A C − B A , [ B , D ] + B A − C B , [ C , D ] + C B − A C is central in ℜ. The universal additive DAHA (double affine Hecke algebra) H of type ( C 1 ∨ , C 1 ) is a unital associative F -algebra generated by t 0 , t 1 , t 0 ∨ , t 1 ∨ and the relations state that t 0 + t 1 + t 0 ∨ + t 1 ∨ = − 1 and each of t 0 2 , t 1 2 , t 0 ∨ 2 , t 1 ∨ 2 is central in H . Each H -module is an ℜ-module by pulling back via the algebra homomorphism ℜ → H given by A ↦ ( t 1 ∨ + t 0 ∨ ) ( t 1 ∨ + t 0 ∨ + 2 ) 4 , B ↦ ( t 1 + t 1 ∨ ) ( t 1 + t 1 ∨ + 2 ) 4 , C ↦ ( t 0 ∨ + t 1 ) ( t 0 ∨ + t 1 + 2 ) 4 . Let V denote any finite-dimensional irreducible H -module. The set of ℜ-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the ℜ-module V is completely reducible if and only if t 0 is diagonalizable on V .

On toric ind-varieties and pro-affine semigroups

An ind-variety is an inductive limit of closed embeddings of algebraic varieties and an ind-group is a group object in the category of ind-varieties. These notions were first introduced by Shafarevich in the study of the automorphism group of affine spaces and have been studied by many authors afterwards. An ind-torus is an ind-group obtained as an inductive limit of closed embeddings of algebraic tori that are also algebraic group homomorphisms. In this paper, we introduce the natural definition of toric ind-varieties as ind-varieties having an ind-torus as an open set and such that the action of the ind-torus on itself by translations extends to a regular action on the whole ind-variety. We are brought to introduce and study pro-affine semigroup that turn out to be unital semigroups isomorphic to closed subsemigroups of the group of arbitrary integer sequences with the product topology such that their projection to the first i-th coordinates is finitely generated for all positive integers i. Our main result is a duality between the categories of affine toric ind-varieties and the category of pro-affine semigroups.