Sum Of Algebras Research Articles

A Lie group investigation of exact weakly nonlinear quasi-planar sound waves

The Lie point symmetries of the (1+2)D Zabolotskaya–Khokholov (ZK) equation for a gas with nonzero viscosity at rest are calculated. They form an infinite‐dimensional Lie algebra which can be written as the direct sum of a 2D Lie algebra and an infinite‐dimensional Lie algebra. The symmetry structure of the ZK equation is used to construct its physically relevant solutions from those of the (1+1)D Burgers equation. In particular, travelling wave solutions (one‐solitons) as well as the multi‐soliton solutions of the ZK equation are obtained. Also, we obtain the N‐waves, the periodic waves and the solution to some Cauchy problems for the ZK equation. We emphasize that our method can be utilized to generate a wider class of solutions of the ZK equation.

Invariants of a semi-direct sum of Lie algebras

We show that any semi-direct sum $L$ of Lie algebras with Levi factor $S$ must be perfect if the representation associated with it does not possess a copy of the trivial representation. As a consequence, all invariant functions of $L$ must be Casimir operators. When $S= \frak{sl}(2,\mathbb{K}),$ the number of invariants is given for all possible dimensions of $L$. Replacing the traditional method of solving the system of determining PDEs by the equivalent problem of solving a system of total differential equations, the invariants are found for all dimensions of the radical up to five. An analysis of the results obtained is made, and this lead to a theorem on invariants of Lie algebras depending only on the elements of certain subalgebras.

Geometry of good sets inn-fold Cartesian product

We propose here a multidimensional generalisation of the notion of link introduced in our previous papers and we discuss some consequences for simplicial measures and sums of function algebras.

LargeNcand chiral dynamics

We study the dependence on the number of colors of the leading $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude in chiral dynamics. We demonstrate the existence of a critical number of colors for and above which the low energy $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude computed from the simple sum of the current algebra and vector meson terms is crossing symmetric and unitary at leading order in a truncated and regularized ${1/N}_{c}$ expansion. The critical number of colors turns out to be ${N}_{c}=6$ and is insensitive to the explicit breaking of chiral symmetry. Below this critical value, an additional state is needed to enforce the unitarity bound; it is a broad one, most likely of a ``four quark'' nature.

On double centralizer properties between quantum groups and Ariki–Koike algebras

In this paper, the double centralizer properties for tensor spaces as bimodules over quantum groups and Ariki–Koike algebras as well as their relations with Shoji's algebra H h (see [J. Algebra 226 (2000) 818–856]) are studied. We proved that, under the separation condition (see [J. Reine Angew. Math. 513 (1999) 53–69]), the natural homomorphism from the Ariki–Koike algebra to the endomorphism algebra of tensor space as module over quantum group is surjective. We also show that the natural homomorphism from Shoji's algebra H h to that endomorphism algebra is always surjective, and H h is actually isomorphic to a direct sum of some matrix algebras over some Hecke algebras of type A.

The predual of [formula omitted]-tensor products over [formula omitted]-subalgebras (separable case)

Let M, N, R be W ∗ -algebras, with R unitally embedded in both M and N. By using Reduction Theory, we describe the predual (M ⊗ ̄ RN) ∗ of the W ∗ -tensor product M ⊗ ̄ RN over the common W ∗ -subalgebra R in the separable predual case. We also analyze the case when R is a direct sum of full matrix algebras, without any separability assumption. In the last situation, the predual (M ⊗ ̄ RN) ∗ is described by the center Z( RM ∗ ⊗N ∗R) of the R– R bimodule RM ∗ ⊗N ∗R , the last one being isomorphic to the predual of M ⊗ ̄ N . It is also shown that such a reduction-free result cannot be extended to the remaining cases.

Contractible Fréchet algebras

A unital Frechet algebra A is called contractible if there exists an element d such that π A (d) = 1 and ad = da for all a ∈ A where π A : A⊗A → A is the canonical Frechet A-bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Frechet algebra A to be a direct sum of a finite-dimensional semisimple algebra M and a contractible Frechet algebra N without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative Imc Frechet Q-algebra is contractible if, and only if, it is algebraically and topologically isomorphic to C n for some n ∈ N. On the other hand, we show that a Frechet algebra, that is, a locally C*-algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.

Primitive Limit Algebras and C*-Envelopes,

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C * -envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C * -envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n -transitive for all n ⩾1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C * -envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal.

Primitive Limit Algebras and C *-Envelopes

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C *-envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C *-envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n-transitive for all n⩾1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C *-envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal.

Approximately Finitely Acting Operator Algebras

Let E be an operator algebra on a Hilbert space with finite-dimensional C*-algebra C*(E). A classification is given of the locally finite algebras A0=[formula](Ak, φk) and the operator algebras A=[formula](Ak, φk) obtained as limits of direct sums of matrix algebras over E with respect to star-extendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup, with scale, which is a right module for the semiring VE=Homu(E⊗K, E⊗K) of unitary equivalence classes of star-extendible homomorphisms. This semigroup is referred to as the dimension module invariant. In the operator algebra case the invariants consist of a metrized additive semigroup with scale and a contractive right module VE-action. Subcategories of algebras determined by restricted classes of embeddings, such as 1-decomposable embeddings between digraph algebras, are also classified in terms of simplified dimension modules.

Numerical Radius Isometries

Let V be a direct sum of full matrix algebras, or the algebra of block upper triangular matrices. Suppose r ( A ) is the numerical radius of $A \\in V$ . We characterize mappings $f: V \\rightarrow V$ that satisfy $r(\\;f(A)-f(B)) = r(A-B) \\ {\\rm for \\ all} \\ A, B \\in V$ .

On the classification of simple inductive limit $C^*$-algebras. I: The reduction theorem

Suppose that A = \lim\limits_{n\to\infty}(A_n = \bigoplus_{i=1}^{t_n} M_{[n,i]}(C(X_{n,i})), \phi_{n,m}) is a simple C^* -algebra, where X_{n,i} are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit C^* -algebras is classified by the Elliott invariant – consisting of the ordered K-group and the tracial state space – in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C^* -algebras in this class do not enjoy the real rank zero property.)

Lie Representations and an Algebra Containing Solomon's

We introduce and study a Hopf algebra containing the descent algebra as a sub-Hopf-algebra. It has the main algebraic properties of the descent algebra, and more: it is a sub-Hopf-algebra of the direct sum of the symmetric group algebras; it is closed under the corresponding inner product; it is cocommutative, so it is an enveloping algebra; it contains all Lie idempotents of the symmetric group algebras. Moreover, its primitive elements are exactly the Lie elements which lie in the symmetric group algebras.

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If $$ {\frak A} $$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of $$ {\frak A} $$ (i) the quotient $$ {\frak A}/L $$ has the approximation property and (ii) the canonical map from $$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp $$ is open, then $$ {\frak A} $$ is finite-dimensional. As an application, we show that, if $${\frak A}$$ is an amenable Banach algebra whose underlying Banach space is an Lp-space with $$ p\in (1,\infty) $$ such that for each maximal left ideal L the quotient $$ {\frak A}/L $$ has the approximation property, then $$ {\frak A} $$ is finite-dimensional.

A CuCh Interpretation of an Object-Oriented Language

Böhm's CuCh machine extends the pure lambda-calculus with algebraic data types and provides the possibility of defining functions over disjoint sums of algebras. We exploit such natural form of overloading to define a functional interpretation of a simple, but significant fragment of a typical objectoriented classbased language.

Orthogonal, symplectic and unitary representations of finite groups

Let K be a field, G a finite group, and ρ : G → GL(V ) a linear representation on the finite dimensional K-space V . The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms h : V × V → K which are G-invariant. II. If h is such a form, enumerate the equivalence classes of representations of G into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on G or K under which two orthogonal, symplectic or unitary representations of G are equivalent if and only if they are equivalent as linear representations and their underlying forms are “isotypically” equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) KG-module components of their spaces are equivalent. We assume throughout that the characteristic of K does not divide 2|G|. Solutions to I and II are given when K is a finite or local field, or when K is a global field and the representation is “split”. The results for III are strongest when the degrees of the absolutely irreducible representations of G are odd – for example if G has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index – and, in the case that K is a local or global field, when the representations are split. Let G be a finite group of order g, K a field of characteristic relatively prime to 2g with an involution (possibly the identity) a 7→ ā, and ρ : G → GL(V ) a linear representation of G on the finite dimensional K-vector space V . The group algebraKG has an involution extending that onK and inverting the group elements. Since KG is a semisimple algebra, the involution algebra (KG, ) is a direct sum of simple involution algebras: (KG, ) = (A1, )⊕ (A2, )⊕ · · · ⊕ (Ar, ), where each Ai either is a simple algebra stabilized by the involution or is a “simple hyperbolic algebra” – a direct sum of two simple algebras interchanged by the involution. Now let

Notes on sums of algebras, subdirect products and subdirectly irreducible algebras

The paper shows that certain sums of algebras admit a representation as subdirect products. As a corollary, one obtains a characterization of subdirectly irreducible algebras. Examples of modes show that the characterization gives a good basis for a detailed investigation of the subdirectly irreducible algebras.

The Structure of Galois Algebras

Let B be a ring with 1 and G an automorphism group of B of order n for some integer n. It is shown that if B is a Galois algebra with Galois group G, then B is either a direct sum of central Galois algebras or a direct sum of central Galois algebras and a commutative Galois algebra. Moreover, when G is inner, B is either a direct sum of Azumaya projective group algebras or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. Examples are given for these structures.

A Continuous Field of Projectionless C*-Algebras

AbstractWe use some results about stable relations to show that some of the simple, stable, projectionless crossed products of O2 by considered by Kishimoto and Kumjian are inductive limits of type C*-algebras. The type I C*-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional C*-algebras.

The Donald–Flanigan Problem for Finite Reflection Groups

The Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types Bn and Dn (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if Sn is the symmetric group then (i) the problem is solvable also for the wreath product H\(\wr\)Sn = H × ··· × H (n times) ⋊ Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H\(\wr\)G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.