Showing papers on "Algebra representation published in 1973"

Linear analysis and representation theory

Abstract: I. Algebras and Banach Algebras.- 1. Algebras and Norms.- 2. The Group of Units and the Quasigroup.- 3. The Maximal Ideal Space.- 4. The Spectrum of an Element.- 5. The Spectral Norm Formula.- 6. Commutative Banach Algebras and their Ideals.- 7. Radical and Semisimplicity.- 8. Involutive Algebras.- 9. H* Algebras.- Remarks.- II. Operators and Operator Algebras.- 1. Topologies on Vector Spaces and on Operator Algebras.- 2. Compact Operators.- 3. The Spectral Theorem for Compact Operators.- 4. Hilbert-Schmidt Operators.- 5. Trace Class Operators.- 6. Vector Valued Line Integrals.- 7. Homomorphisms into A. The Spectral Mapping Theorem.- 8. Unbounded Operators.- Remarks.- III. The Spectral Theorem, Stable Subspaces and v. Neumann Algebras.- 1. Linear Functionals on Vector Lattices and their Extensions.- 2. Linear Functionals on Lattices of Functions.- 3. The Spectral Theorem for SelfAdjoint Operators in Hilbert Space.- 4. Normal Elements and Normal Operators.- 5. Stable Subspaces and Commutants.- 6. von Neumann Algebras.- 7. Measures on Locally Compact Spaces.- Remarks.- IV. Elementary Representation Theory in Hilbert Space.- 1. Representations and Morphisms.- 2. Irreducible Components, Equivalence.- 3. Intertwining Operators.- 4. Schur's Lemma.- 5. Multiplicity of Irreducible Components.- 6. The General Trace Formula.- 7. Primary Representations and Factorial v. Neumann Algebras.- 8. Algebras and Representations of Type I.- 9. Type II and III v. Neumann Algebras.- Remarks.- Preliminary Remarks to Chapter V.- V. Topological Groups, Invariant Measures, Convolutions and Representations.- 1. Topological Groups and Homogeneous Spaces.- 2. Haar Measure.- 3. Quasi-Invariant and Relatively Invariant Measures.- 4. Convolutions of Functions and Measures.- 5. The Algebra Representation Associated with ?:S??(?).- 6. The Regular Representations of Locally Compact Groups.- 7. Continuity of Group Representations and the Gelfand-Raikov Theorem.- Remarks.- VI. Induced Representations.- 1. The Riesz-Fischer Theorem.- 2. Induced Representations when G/H has an Invariant Measure.- 3. Tensor Products.- 4. Induced Representations for Arbitrary G and H.- 5. The Existence ofa Kernel for L1(G)??(K).- 6. The Direct Sum Decomposition of the Induced Representation ?:G?u(K).- 7. The Isometric Isomorphism between ?2 and HS(K2, K1). The Computation of the Trace in Terms of the Associated Kernel.- 8. The Tensor Product of Induced Representations.- 9. The Theorem on Induction in Stages.- 10. Representations Induced by Representations of Conjugate Subgroups.- 11. Mackey's Theorem on Strong Intertwining Numbers and Some of its Consequences.- 12. Isomorphism Theorems Implying the Frobenius Reciprocity Relation.- Remarks.- VII. Square Integrable Representations, Spherical Functions and Trace Formulas.- 1. Square Integrable Representations and the Representation Theory of Compact Groups.- 2. Zonal Spherical Functions.- 3. Spherical Functions of Arbitrary Type and Height.- 4. Godement's Theorem on the Characterization of Spherical Functions.- 5. Representations of Groups with an Iwasawa Decomposition.- 6. Trace Formulas.- Remarks.- VIII. Lie Algebras, Manifolds and Lie Groups.- 1. Lie Algebras.- 2. Finite Dimensional Representations of Lie Algebras. Cartan's Criteria and the Theorems of Engel and Lie.- 3. Presheaves and Sheaves.- 4. Differentiable Manifolds.- 5. Lie Groups and their Lie Algebras.- 6. The Exponential Map and Canonical Coordinates.- 7. Lie Subgroups and Subalgebras.- 8. Invariant Lie Subgroups and Quotients of Lie Groups. The Projective Groups and the Lorentz Group.- Remarks.- Index of Notations and Special Symbols.

The polynomial identities of the Grassmann algebra

Abstract: By using the theory of codimensions the ¿\"-ideal of polynomial identities of the Grassmann (exterior) algebra is computed.

About the non-relativistic structure of the conformal algebra

Abstract: A new basis of the conformal algebra is proposed, which makes appear two conjugated Schrodinger algebras. This basis allows to exhibit a chain which does not contain the Poincare algebra, between theI $$\mathcal{O}$$ (4, 2) algebra and the (two-dimensional) extended Galilean one. This non-relativistic structure of the conformal algebra is well adapted to discuss some extreme models of hadrons based on collinear massless particles.

Spectrum-Generating Algebra and No-Ghost Theorem for the Neveu-Schwarz Model

Abstract: The no-ghost theorem of Brower for the conventional dual model (intercept ${\ensuremath{\alpha}}_{0}=1$) is extended to the Neveu-Schwarz model, and its spectrum-generating algebra is constructed. These conformally invariant generators (${A}_{n}, {B}_{s}$) are obtained by a systematic use of the gauges ${G}_{r}$. As a consequence it is demonstrated that particular dual models having resonances with positive-definite norms and masses are possible.

On the Algebra of Test Functions for Field Operators Decomposition of Linear Functionals into Positive Ones

Abstract: It is shown that every continuous linear functional on the field algebra can be defined by a vector in the Hilbert space of some representation of the algebra. The functionals which can be written as a difference of positive ones are characterized. By an example it is shown that a positive functional on a subalgebra does not always have an extension to a positive functional on the whole algebra.

Parastatistics and the quark model

Abstract: An extension of ordinary parastatistics is considered which makes use of all the representations of the parastatistics algebra obtained from the usual ansatz. Govorkov's demonstration that such an extension, for parastatistics of order 2, implies a U(2) symmetry, is generalized for parastatistics of order p. The parastatistics algebra, restricted to N dynamical states, is characterized by the irreducible representations of U(N), S O(2N), and S O(2N + 1) which it contains. It is shown that these representations have multiplicities equal to the dimensions of associated representations of U(p), O(p) and C(p), respectively, where C(p) is a subalgebra of the enveloping algebra of O(p), but is not a Lie algebra. The symmetric group S(p) also appears, as a subalgebra of the enveloping algebra of C(p). It is shown how a nondegenerate vacuum state may be defined for the generalized parastatistics algebra of order p, and how to construct state vectors corresponding to arbitrary numbers of quarklike particles and antiparticles. Such states belong to irreducible representations of U(N), and can be obtained by the application of one kind of creation and annihilation operators to certain basic states, here called reservoir states, which correspond to the different irreducible representations of S O(2N + 1). The specialization to parastatistics of order 3 is discussed in detail with the application to a quark model of the hadrons in view. It is shown how to define isospin and hypercharge in a significant way in this model, which, however, differs in some respects from Gell‐Mann's well‐known 3‐fermion model, and also from Greenberg's 3‐parafermion model. Some of the physical implications are examined.

Representation of $A$-convex algebras

Abstract: Algebraic properties of A-convex algebras are developed via a functor to locally m-convex algebras. The Gel'fandMazur theorem holds for A-convex algebras, and this fact allows a Gel'fand-type representation theorem for a subclass of uniformly A-convex algebras. Connections to existing functional representation theory are also obtained.

On Some Varieties of Associative Algebras

Abstract: A general method is given for proving that certain T-ideals are Spechtian. In particular, the T-ideal is shown to be Spechtian.

On some infinitely presented associative algebras

Abstract: We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Real homogeneous algebras

Abstract: Let (A, ,u) be a finite dimensional real algebra (not necessarily associative) with multiplication isgO. Assuming that Aut(A) is transitive on one-dimensional subspaces we determine all such algebras. There are up to isomorphism only four such algebras, one in each of the dimensions 1, 3, 6, 7. Introduction. For the terminology we refer to Bourbaki [3]. All the algebras considered in this paper are assumed to be finite dimensional. Let A be an algebra over a field F, ,u:A0A--A its multiplication and Aut(A) the group of algebra automorphisms of A. We shall say that A is homogeneous if Aut(A) is transitive on one-dimensional subspaces of A, we shall say that A is extremely homogeneous if Aut(A) is transitive on A\{O}. Kostrikin [7] has shown that if char F#2, A extremely homogeneous and go$0, then F must be a finite field. On the other hand Shult [11] has shown that if A is homogeneous, F=GF(q), q>2 and 4u#O then A_F. The case F=GF(2) has been considered by Gross [4]. Swierczkowski has shown [13] that when F=R (the real field) and A is a homogeneous Lie algebra with ue0 then A is isomorphic to the Lie algebra of skew-symmetric 3 x 3 real matrices. Many of these results have been improved by Mr. L. Sweet [12]. In particular, he has determined all two-dimensional homogeneous algebras and has shown that there are no nontrivial homogeneous algebras over an algebraically closed field. In this paper we shall determine all real homogeneous algebras. If A is an F-algebra and B cA a subspace we define a multiplication in B by choosing a vector space complement C for B in A and putting FuB(bl 0 b2) = i/yA(bl ? b2) where 77: A--B is the projection with kernel C. We say then that (B, FB) is obtained from (A, ,u) by truncation. Note that the definition of lB depends on the choice of C. Received by the editors July 10, 1972. AMS (MOS) subject classqifcations (1970). Primary 17A99; Secondary 17B10, 20G15.

Algebra of observables and quantum logic

Abstract: © Gauthier-Villars, 1973, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Deformations of inhomogeneous classical Lie algebras to the algebras of the linear groups

Abstract: We study a new class of deformations of algebra representations, namely, i2so(n)⇒sl(n,R), i2u(n)⇒sl(n,C)⊕u(1) and i2sp(n)⊕sp(1)⇒sl(n,Q)⊕sp(1). The new generators are built as commutators between the Casimir invariant of the maximal compact subalgebra and a second‐rank mixed tensor. These algebra deformations are related to multiplier representations and manifold mappings of the corresponding Lie groups. Behavior of the representations under Inonu‐Wigner contractions is exhibited. Through the use of these methods we can construct a principal degenerate series of representations of the linear groups and their algebras.

On the equivalence of twisted group algebras and Banach *-algebraic bundles

Abstract: The relationship between twisted group algebras and Banach *-algebraic bundles is investigated. Informally stated, the results are that bundles with Borel cross sections correspond to twisted group algebras, and "locally continuous" twisted group algebras correspond to bundles. In the separable case, these results combine to give a complete correspondence between the bundles and the "locally continuous" algebras. Fell, in [5], developed the theory of Banach *-algebraic bundles over a locally compact group. He was able to treat group extensions, the covariance algebras of quantum mechanics, the transformation group C*-algebras of Glimm (see [6]) and Effros and Hahn (see [4]), and other examples as special cases, and he extended the Mackey-Blattner theory of induced representations to this general setting. On the other hand, Horst Leptin introduced the concept of generalized L1-algebra in [7] and this concept was further developed in [1] under the name twisted group algebra. All the above mentioned examples can be treated in the context of twisted group algebras, and this has led to speculation concerning the precise relationship betweer. these algebras and Fell's bundles. This problem has been mentioned by Fell in [5] and Leptin in [8]. In this note we will establish, in a rather simple way, much of this relationship. We need to point out here that the theory of twisted group algebras developed in [1] is restricted to the case of separable object algebra and second countable group. These restrictions are largely for convenience and are not used essentially until late in that paper. It is easy to see that the basic definitions and results are valid without the countability restrictions, and in fact Leptin develops completely analogous objects without such restrictions. Thus we have not hesitated to state our results in a nonseparable context, and we refer the reader to [7] for the techniques of generalization. We will give a brief description of those parts of the above theories which we will use. We refer the reader to [1] and [5] for any definitions, Received by the editors May 24, 1971 and, in revised form, March 20, 1972 and May 2, 1972. AMS (MOS) subject classifications (1969). Primary 4680; Secondary 4650, 4660.