The Graded Structure of Nondegenerate Meson Algebras

Meson algebras are involved in the wave equation of meson particles in the same way as Clifford algebras are involved in the Dirac wave equation of electrons. Here we improve and generalize the information already obtained about their structure and their representations, when the symmetric bilinear form under consideration is nondegenerate. We emphasize their parity grading. We calculate the center of these meson algebras, and the center of their even subalgebra. Finally we show that every nondegenerate meson algebra over a field contains a group isomorphic to the group of automorphisms of the symmetric bilinear form.      

Clifford Algebras of Forms of Higher Degrees

The aim of this lecture is to introduce Clifford algebras of polynomial forms of higher degrees. We recall that these algebras are in general of infinite dimension, and we give a basis depending on a given basis of the underlying vector space. We then show that, though they contain large free associative algebras, we may construct finite dimensional representations of these algebras, also called linearizations of the polynomial form. If the polynomial form is, in a certain sense, non degenerate, the dimensions of these representations are multiples of the degree of the form. In the end, we recall some results known for the special case of a binary cubic form with at least one simple zero, when explicit computations can be done: the Clifford algebra is an Azumaya algebra of rank 9 over its center, which is the algebra of functions over a cubic curve depending on the given cubic form.     

Interior Multiplications and Deformations with Meson Algebras

A meson algebra is involved in the Duffin wave equation for mesons in the same way as a Clifford algebra is involved in the Dirac wave equation for electrons. Therefore meson algebras too should have geometrical properties after the manner of Grassmann. Actually it is possible to define interior multiplications with similar properties, and deformations too. Every meson algebra is a deformation of a neutral meson algebra, in the same way as (almost) every Clifford algebra is a deformation of an exterior algebra. Some applications follow: the PBW-property is proved for all meson algebras, the injectiveness of Jacobson’s diagonal morphism is proved with the minimal hypothesis, and the existence of Lipschitz monoids is established at least for meson algebras over fields.      

Cohomological reduction by split pairs

A new method is developed to compare cohomology in module categories of different rings. This method does in general not produce isomorphisms, but surjective (or injective) maps between extension groups of modules over the two rings involved. Applications of this method are given to abstract problems—we recover and extend results on the strong no loops conjecture—and to algebras naturally coming up in invariant theory—we relate the cohomology of Brauer algebras with that of various symmetric groups.     

The Clifford algebra of differential forms

This paper reviews Clifford algebras in mathematics and in theoretical physics. In particular, the little-known differential form realization is constructed in detail for the four-dimensional Minkowski space. This setting is then used to describe spinors as differential forms, and to solve the Klein-Gordon and Kähler-Dirac equations. The approach of this paper, in obtaining the solutions directly in terms of differential forms, is much more elegant and concise than the traditional explicit matrix methods. A theorem given here differentiates between the two real forms of the Dirac algebra by showing that spin can be accommodated in only one of them.     

The design of linear algebra and geometry

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.     

Clifford-algebraic random walks on the hypercube

The n-dimensional hypercube is a simple graph on 2ⁿ vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing.     

Extended Grassmann and Clifford Algebras

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras can be embedded in which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms. R. da Rocha is supported by CAPES     

Cyclotomic Solomon algebras

This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type G(r,1,n). As with the Solomon descent algebra, our algebra has a basis given by sums of ‘distinguished’ coset representatives for certain ‘reflection subgroups.’ We explicitly describe the structure constants with respect to this basis and show that they are polynomials in r. This allows us to define a deformation, or q-analogue, of these algebras which depends on a parameter q. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.     

Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field. Received: 29 April 1999     

Skew Clifford algebras

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the ${\mathbb{Z}}_2$-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.     

On a class of algebras associated to directed graphs

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over noncommutative algebras. We also construct a basis for our algebras associated to layered graphs.     

Finitely presented algebras and groups defined by permutation relations

The class of finitely presented algebras over a field K with a set of generators a₁,…,a_n and defined by homogeneous relations of the form a₁a₂?a_n=a_σ(1)a_σ(2)?a_σ(n), where σ runs through a subset H of the symmetric group Sym_n of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Sym_n of order n. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Sym_n are proposed.     

Isotriviality and étale cohomology of Laurent polynomial rings

We provide a detailed study of torsors over Laurent polynomial rings under the action of an algebraic group. As applications we obtained variations of Raghunathan’s results on torsors over affine space, isotriviality results for reductive group schemes and forms of algebras, and decomposition properties for Azumaya algebras.     

Conjugation in Brauer algebras and applications to character theory

The Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.     

Unitary Spaces on Clifford Algebras

For the complex Clifford algebra (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These representations take into account the structure of unitary space on Clifford algebra. The work of N.M. is supported in part by the Russian President’s grant NSh-6705.2006.1.     

Skew quasisymmetric Schur functions and noncommutative Schur functions

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.     

Cartan determinants for gentle algebras

The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.Received: 29 October 2004