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Complex-valued Hopfield neural networks have been extended to several high-dimensional Hopfield neural networks (HDHNNs) using hypercomplex numbers. However, the extensions by hypercomplex numbers are limited. For example, the dimensions of Clifford and Cayley-Dickson algebras are power of two. Group theory provides more various algebras since the orders of groups are any. In this work, Klein four-group is introduced to HDHNNs. It is an abelian group whose order is 4. A Klein Hopfield neural network employs the twin-multistate activation function. We compare the noise tolerance with that of quaternion-valued and commutative quaternion-valued twin-multistate Hopfield neural networks by computer simulations. [ABSTRACT FROM AUTHOR]

In a group G , elements a and b are conjugate if there exists g ∈ G such that g − 1 a g = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S , a is conjugate to b , which we will write as a ∼ i b , if there exists g ∈ S 1 such that g − 1 a g = b and g b g − 1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P -semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups. [ABSTRACT FROM AUTHOR]

A semigroup is called a GV-inverse semigroup if and only if it is isomorphic to a semi-lattice of π-groups. In this paper, we give the suficient and necessary conditions for a GV-inverse semigroup to be a strong semilattice of π-groups. Some conclusions about Clifford semigroups are generalized. [ABSTRACT FROM AUTHOR]

The q-qubit Clifford group, that is, the normalizer of the q-qubit Pauli group in U(2q), is a fundamental structure in quantum information with a wide variety of applications. We characterize all irreducible subrepresentations of the two-copy representation φ⊗2 of the Clifford group on the two-fold tensor product of the space of linear operators M 2 q ⊗ 2 . In the companion paper [Helsen et al., e-print arXiv:1701.04299 (2017)], we apply this result to improve the statistics of randomized benchmarking, a method for characterizing quantum systems. [ABSTRACT FROM AUTHOR]

Takahasi's Theoremon chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. [ABSTRACT FROM AUTHOR]

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing. [ABSTRACT FROM AUTHOR]

In this paper, we establish a strong connection between groups and gyrogroups, which provides the machinery for studying gyrogroups via group theory. Specifically, we prove that there is a correspondence between the class of gyrogroups and a class of triples with components being groups and twisted subgroups. This in particular provides a construction of a gyrogroup from a group with an automorphism of order two that satisfies the uniquely 2-divisible property. We then present various examples of such groups, including the general linear groups over and , the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously. [ABSTRACT FROM AUTHOR]

The representation theory of 0-Hecke–Clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning. Bergeron, Hivert and Thibon have proved that the Grothendieck ring of the category of finitely generated supermodules of 0-Hecke–Clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by Stembridge. In this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups. In particular, two kinds of restriction rules for induced projective supermodules are obtained. After that, we consider the corresponding Heisenberg double and its Fock representation to prove that the ring of peak quasisymmetric functions is free over its subring of symmetric functions spanned by Schur's Q-functions. [ABSTRACT FROM AUTHOR]

We determine the gonality and the Clifford index for curves on a compact smooth toric surface. Moreover, it is shown that their gonality is computed by pencils on the ambient surface. From the geometrical viewpoint, this means that the gonality can be read off from the lattice polygon associated to the curve. [ABSTRACT FROM AUTHOR]

Using the Clifford algebra formalism, we give an algebraic proof that the open unit ball B = { v ∈ R n : ‖ v ‖ < 1 } of R n equipped with Einstein addition ⊕ E forms a B -loop or, equivalently, a uniquely 2-divisible gyrocommutative gyrogroup. We obtain a compact formula for Einstein addition in terms of Möbius addition. We then give a characterization of associativity and commutativity of vectors in B with respect to Einstein addition. [ABSTRACT FROM AUTHOR]

We study presentations, defined by Sidki, resulting in groups y ( m , n ) that are conjectured to be finite orthogonal groups of dimension m + 1 in characteristic two. This conjecture, if true, shows an interesting pattern, possibly connected with Bott periodicity. It would also give new presentations for a large family of finite orthogonal groups in characteristic two, with no generator having the same order as the cyclic group of the field. We generalise the presentation to an infinite version y ( m ) and explicitly relate this to previous work done by Sidki. The original groups y ( m , n ) can be found as quotients over congruence subgroups of y ( m ) . We give two representations of our group y ( m ) . One into an orthogonal group of dimension m + 1 and the other, using Clifford algebras, into the corresponding pin group, both defined over a ring in characteristic two. Hence, this gives two different actions of the group. Sidki's homomorphism into SL 2 m − 2 ( R ) is recovered and extended as an action on a submodule of the Clifford algebra. [ABSTRACT FROM AUTHOR]

The group 27:Sp6(2) is a maximal subgroup ofAut(Fi22) of index 694980. The groupAut(Fi22) is the full automorphism group of the smallest Fischer simple groupFi22. In this paper, we construct the character table of the inertia group 27:(25:S6) of 27:Sp6(2) of index 63 through its Fischer-Clifford matrices. The split extension group 27:(25:S6) sits maximally inside 27:Sp6(2). [ABSTRACT FROM PUBLISHER]

We introduce the reader to Clifford theory; i. e. how to obtain integral (p-adic) representations of the group G from representations of a normal subgroup, whose order is a unit, in the base ring. We formulate a version of Clifford theory for automorphisms and apply this to construct. a group and a non inner automorphism of it, which becomes inner in an integral group-ring. This is then used to show that there are two (infinite) non-isomorphic poly-cyclic groups, which have isomorphic integral group-rings. [ABSTRACT FROM AUTHOR]

We give an algorithm which produces a unique element of the Clifford group on n qubits (Cn) from an integer 0 ≤ i < |Cn| (the number of elements in thegroup). The algorithm involves O(n³) operations and provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of Cn which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n³). [ABSTRACT FROM AUTHOR]

By the use of condition of relativistic covariance, Dirac group theory, Clifford algebra and complete orthonormal sets of ψ(α*)-self-frictional exponential type orbitals (ψ(α*)-S FETOs) introduced by the author in standard convention, the Hartree-Fock (HF) theory is suggested for multideterminantal single configuration states with any number of open shells of atoms and molecules constructed from the Standard Model-Fermi (SM-F) particles with m≠0,s = 0, and e = 0 defined in the Standard Model of particle physics. It is shown that the origin of stability of these systems is the quantum damping or self-frictional forces produced by the SM-F particle itself. As an application, we have presented the periodic table for the SM-F atomic elements using Pauli principle of spinless noncharged identical SM-F particles. [ABSTRACT FROM AUTHOR]

In this paper, we study Clifford-Wolf translations of Finsler spaces. We give a characterization of those Clifford-Wolf translations generated by Killing vector fields. In particular, we show that there is a natural interrelation between the local one-parameter groups of Clifford-Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for a Killing vector field to have a constant length, in which case the local one-parameter group of isometries generated by the Killing field consist of Clifford-Wolf translations. Finally, we construct explicit examples to explain some of the results of this paper. [ABSTRACT FROM AUTHOR]

Representations of SO(4, 2) are constructed using 4 × 4 and 2 × 2 matrices with elements in H'⊗C and the known isomorphism between the conformal group and SO(4,2) is written explicitly in terms of the 4 × 4 representation. The Clifford algebra structure of SO(4, 2) is briefly discussed in this language, as is its relationship to other groups of physical interest. [ABSTRACT FROM AUTHOR]

A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their nontrivial second cohomology classes, which give rise to their central extensions (the affine Kac–Moody groups and Lie algebras). Loop groups embed into the group of continuous automorphisms of , and these classes come from a second cohomology class of . In a similar way, double loop groups embed into a group of automorphisms of , denoted by , which has a nontrivial third cohomology. In this paper, we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them “gerbal representations”. We then construct a gerbal representation of (and hence of double loop groups), realizing its nontrivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analog of the fermionic Fock representations of the ordinary loop groups. [ABSTRACT FROM AUTHOR]

In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Alg. (to appear)] a signature ϵ = (p, q)-dependent transposition anti-involution of real Clifford algebras Cℓ p,q for non-degenerate quadratic forms was introduced. In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Alg. (to appear)] we showed that, depending on the value of (p − q) mod 8, the map gives rise to transposition, complex Hermitian or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [R. Abłamowicz and B. Fauser, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Comput. Phys. Commun. 170 (2005), pp. 115–130]. We provide a full signature (p, q)-dependent classification of the invariance groups of this product for p + q ≤ 9. The map is identified as the ‘star’ map known [D.S. Passman, The Algebraic Structure of Group Rings, Robert E. Krieger Publishing Company, Malabar, Florida, 1985] from the theory of (twisted) group algebras where the Clifford algebra Cℓ p,q is seen as a twisted group ring n = p + q. We discuss important subgroups of a stabilizer group G p,q (f) of a primitive idempotent f and we relate their transversals to spinor bases in spinor spaces realized as minimal left ideals Cℓ p,q  f. [ABSTRACT FROM PUBLISHER]

CPT groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincaré group which describe orientable (extended) objects. A general method for construction of CPT groups of the fields of any spin is given. CPT groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. CPT groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of the same spin with different masses. [ABSTRACT FROM AUTHOR]

We prove that a countably compact Clifford topological semigroup S is metrizable if and only if the set E={ e∈ S: ee= e} of idempotents of S is a metrizable G-set in S. [ABSTRACT FROM AUTHOR]

In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator DΔk/2 for some even k ∈ Z. They will be called k-holomorphic Cliffordian automorphic forms. k-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. This function class includes the set of k-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing k-hypermonogenic cusp forms for k ≠ 0, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms, we turn to the construction of Poincaré series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of k-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and of k-holomorphic Cliffordian cusp forms. [ABSTRACT FROM AUTHOR]

In this paper, we give a new framework for constructing low ML decoding complexity space-time block codes (STBCs) using codes over the Klein group \cal K. Almost all known low ML decoding complexity STBCs can be obtained via this approach. New full-diversity STBCs with low ML decoding complexity and cubic shaping property are constructed, via codes over \cal K, for number of transmit antennas N=2^m, m\geq 1, and rates R>1 complex symbols per channel use. When R=N, the new STBCs are information-lossless as well. The new class of STBCs have the least known ML decoding complexity among all the codes available in the literature for a large set of (N,R) pairs. [ABSTRACT FROM AUTHOR]

In the first article [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Algebra, to appear] we showed that real Clifford algebras Cℓ(V, Q) possess a unique transposition anti-involution . The map reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of associated matrix of that element in the left regular representation of the algebra. In this article we show that, depending on the value of (p − q) mod 8, where ϵ = (p, q) is the signature of Q, the anti-involution gives rise to transposition, Hermitian complex and Hermitian quaternionic conjugation of representation matrices in spinor representations. We realize spinors in minimal left ideals S = Cℓ p,q  f generated by a primitive idempotent f. The map allows us to define a dual spinor space S*, and a new spinor norm on S, which is different, in general, from two spinor norms known to exist. We study a transitive action of generalized Salingaros' multiplicative vee groups G p,q on complete sets of mutually annihilating primitive idempotents. Using the normal stabilizer subgroup G p,q (f), we construct left transversals, spinor bases and maps between spinor spaces for different orthogonal idempotents f i summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases. [ABSTRACT FROM AUTHOR]

Abstract: Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper, we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for fermionic modes, which can be implemented by means of unitary transformations, is isomorphic to . The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra ] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington’s -numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions . From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonians. Relying on Jordan-Wigner transformations for two-spin- and single-spin- systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin- and single-spin- Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of and matrices (of respective sizes 4×4 and 6×6) and their mutual relation. [Copyright &y& Elsevier]

The notion of G-algebra equivalence for a group G is generalized from the setting of G-algebras over fields to G-algebras over commutative rings. This leads to a formulation of Turull's Brauer–Clifford group for separable G-algebras over commutative rings, and to connections with Fröhlich and Wall's equivariant Brauer group. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the pseudo-amenability of semigroup algebra ℓ( S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup $S={\mathcal{M}}^{0}(G,I)$, the pseudo-amenability of ℓ( S) is equivalent to the amenability of G. [ABSTRACT FROM AUTHOR]

In this article, we investigate a problem that was proposed by Petrich and Reilly in [2]: What can be said about Y*? We get that Y* ∈ [ε, ν] on completely regular semigroups. Furthermore, we give a description of Y* on completely simple semigroups and normal cryptogroups, respectively. [ABSTRACT FROM AUTHOR]

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω- E-reflexive semigroups, are studied. (( ω)) and ((( ω))) are found to be the least Eω-Clifford and Eω- E-reflexive congruences on S, respectively. Characterizations of the congruences (( ω)) and ((( ω))) are developed. The lattices of all Eω-Clifford and Eω- E-reflexive congruences are also studied. [ABSTRACT FROM AUTHOR]

Quantum teleportation is a computational primitive that allows non-local quantum communication and quantum computation. In this work, we present two schemes for quantum gate teleportation. The first scheme shows under what conditions an n-qudit gate can be teleported using a generalization of Gottesman-Chuang procedure [Nature 402, 390 (1999)]. The second scheme shows that quantum gate teleportation can be transformed in the teleportation of a single-qudit. [ABSTRACT FROM AUTHOR]

Abstract: We present an algorithmic proof of the theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan–Dieudonné–Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation. [ABSTRACT FROM AUTHOR]

In this paper, we give a self-contained proof of the open problem 퓞퓑퓖 ∨ 퓑퓐 = 퓒퓑퓖 proposed by Petrich and Reilly in 1999 [ABSTRACT FROM AUTHOR]

Abstract: Using the Clifford algebra formalism we study the Möbius gyrogroup of the ball of radius t of the paravector space , where V is a finite-dimensional real vector space. We characterize all the gyro-subgroups of the Möbius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyro-subgroup for the paravector ball. The geometric and algebraic properties of the equivalence classes are investigated. We show that the equivalence classes locate in a k-dimensional sphere, where k is the dimension of the gyro-subgroup, and the resulting quotient spaces are again Möbius gyrogroups. With the algebraic structure of the factorizations we study the sections of Möbius fiber bundles inherited by the Möbius projectors. [ABSTRACT FROM AUTHOR]

A graded tensor category over a group G will be called a strongly G-graded tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the module categories over a strongly G-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of G. [ABSTRACT FROM PUBLISHER]

Let N be a normal subgroup of a finite group G such that b is a block of defect zero on N. (This will be the case for any block of N if N is a p′-group.) Let M be an indecomposable G-module with vertex Q that lies in a block of G that covers b. We present a procedure for obtaining a Green correspondent of M relative to , , and for obtaining Q-sources of M using Clifford-theoretic analyses. [ABSTRACT FROM AUTHOR]

Let N be a normal subgroup of the finite group X and let G = X/ N. It is well known that the number k( X) = |Irr( X)| of conjugacy classes (irreducible characters) of X is bounded above by the product k( N) k( G). For more precise results one has to determine, for any θ ∈ Irr( N), the number of G-conjugates of θ and the number kθ( G) of irreducible characters of X lying above θ. Clifford theory reduces the computation of kθ( G) to the situation where N = Z is central in X, and then it only depends on the Clifford obstruction μ = μG( θ). The class number kμ( G) is defined for any μ ∈ H2( G, ℂ*), and behaves well when passing to isoclinic central extensions. Suppose that μ ≠ 1. There are examples where kμ( G) = 1, in which case X is a group of central type, and examples where kμ( G) = k( G) is as large as possible. In the first case G must be solvable (Howlett–Isaacs). The latter case can happen when G is perfect but not when G is a simple or, more generally, a quasisimple group. [ABSTRACT FROM AUTHOR]

A candidate action for an Exceptional E8 gauge theory of gravity in 8D is constructed. It is obtained by recasting the E8 group as the semi-direct product of GL(8,R) with a deformed Weyl–Heisenberg group associated with canonical-conjugate pairs of vectorial and antisymmetric tensorial generators of rank two and three. Other actions are proposed, like the quarticE8 group-invariant action in 8D associated with the Chern–Simons E8 gauge theory defined on the 7-dim boundary of a 8D bulk. To finalize, it is shown how the E8 gauge theory of gravity can be embedded into a more general extended gravitational theory in Clifford spaces associated with the Cl(16) algebra and providing a solid geometrical program of a grand unification of gravity with Yang–Mills theories. The key question remains if this novel gravitational model based on gauging the E8 group may still be renormalizable without spoiling unitarity at the quantum level. [ABSTRACT FROM AUTHOR]

The article deals with Schur indices which is not divisible by p, and the finite group G is either supersolvable or p-nilpotent. The author believes that it is more a question of Clifford theory than of modular representation theory, based on various propositions in which the supersolvable group is a group all of whose chief factors are cyclic. An example of a solvable group G is presented.

We show that G-graded Rickard equivalences defined over small fields preserve Clifford classes associated to characters. These equivalences are compatible with operation on Clifford classes defined in terms of central simple crossed products. [ABSTRACT FROM AUTHOR]

This is an introduction to Clifford theory of induced representations from normal subgroups of finite groups. As an application, a complete explicit description of the irreducible representations of a class of metacyclic groups and of the groups with a subgroup N of index two (in terms of the representations of N) is given. [ABSTRACT FROM AUTHOR]

A regular orthogroup S with the property that D e = R e or D e = L e for any idempotent e∈ S is called a WLR-regular orthogroup. In this paper, we give constructions of such semigroups in terms of spined products of left and right regular orthogroups with respect to Clifford semigroups. WLR-cryptogroups and its special cases are also investigated. [ABSTRACT FROM AUTHOR]

We investigate a theory in which fundamental objects are branes described in terms of higher grade coordinates $$X^{\mu{_1}\ldots \mu{_n}}$$ encoding both the motion of a brane as a whole, and its volume evolution. We thus formulate a dynamics which generalizes the dynamics of the usual branes. Geometrically, coordinates $$X^{\mu{_1} \ldots \mu{_n}}$$ and associated coordinate frame fields { $${\gamma_{\mu{_1}\ldots\mu{_n}}}$$ } extend the notion of geometry from spacetime to that of an enlarged space, called Clifford space or C-space. If we start from four-dimensional spacetime, then the dimension of C-space is 16. The fact that C-space has more than four dimensions suggests that it could serve as a realization of Kaluza-Klein idea. The “extra dimensions” are not just the ordinary extra dimensions, they are related to the volume degrees of freedom, therefore they are physical, and need not be compactified. Gauge fields are due to the metric of Clifford space. It turns out that amongst the latter gauge fields there also exist higher grade, antisymmetric fields of the Kalb–Ramond type, and their non-Abelian generalization. All those fields are naturally coupled to the generalized branes, whose dynamics is given by a generalized Howe–Tucker action in curved C-space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definition of groupoid graded rings. Group graded rings, (skew) groupoid rings, artinian semisimple rings, matrix rings and others can be regarded as special kinds of groupoid graded rings. Our main task is to classify strongly groupoid graded rings by cohomology of groupoids. Some classical results about group graded rings are generalized to groupoid graded rings. In particular, the Clifford Theorem for a strongly groupoid graded ring is given. [ABSTRACT FROM AUTHOR]

Clifford Theory concerns the representations over a field F of a finite group G and one of the normal subgroups N of G. It is often viewed as a series of reduction theorems. Clifford classes are certain equivalence classes of G/N-algebras over F. They have been used to describe the Schur indices of the irreducible characters of certain families of classical groups. In the present paper, we show how Clifford classes can also be used to explain and to reflect the reduction theorems of classical Clifford theory. We show that certain products of characters correspond to certain products of Clifford classes. We show that induction, restriction, inflation, and extension of the base field can all be defined naturally for Clifford classes. We prove that the properties of these elementary operations on Clifford classes imply all of the standard Clifford-theoretic reductions in the case when the field is ℂ. They also provide important information in the case when the field F is an arbitrary field of characteristic zero. [ABSTRACT FROM AUTHOR]

For finite groups G, it is shown that central simple G-algebras arise naturally from irreducible modules of twisted group algebras over fields of characteristic 0. This leads to an extension of Turull's Clifford theory with Schur indices that applies in the setting of projective representations of finite groups. This theory has the same properties as in the ordinary representation theory setting, perhaps the most interesting of which is a Schur index preserving bijection between certain sets of irreducible projective characters. [ABSTRACT FROM AUTHOR]

A kind of relational mappings, generated from usual set-valued mappings by adding some relative equivalences, is introduced in this paper. With this interesting and powerful tool, one can conveniently discuss some classes of orthogroups and some more general classes of semigroups in the scope of semisuperabundant semigroups. Relational homomorphisms over some congruences on semigroups are used to clarify the definition of a refined semilattice of semigroups. With this notion we investigate the properties and describe the structure of regular orthocryptou semigroups. Such description can also be specialized to various subclasses of regular orthocryptou semigroups, such as left quasi-normal orthocryptou semigroups, left regular orthocryptou semigroups, left C-rpp semi-groups in which Η is a congruence, regular orthocryptogroups, left quasi-normal orthocryptogroups, left regular orthocryptogroups, Clifford semigroups, regular bands, left quasi-normal bands and left regular bands. [ABSTRACT FROM AUTHOR]

Regular semigroups S with the property eS c Se or Se c eS for all idempotents e ∈ S include all left and right Clifford semigroups. Characterizations of such semigroups are given and their structure investigated, in particular in terms of spined products of left and right Clifford semigroups with respect to Clifford semigroups. [ABSTRACT FROM AUTHOR]

We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators encountered in the context of Dirac algebra. Then we show how a recent general mathematical definition of chirality unifies the chirality concepts and resolve conflicting conclusions about symmetry operators, and particularly about the so-called chirality operator. The proofs are based on group theory rather than on Clifford algebras. The results are independent on the representations of Dirac gamma matrices, and stand for higher dimensional ones. [ABSTRACT FROM AUTHOR]