arXiv subjects

Bertfried Fauser

Publications and source records attributed to Bertfried Fauser.

At least 19 recordsLinked to original sources

arXiv2016

Plethystic Vertex Operators and Boson-Fermion Correspondences

We study the algebraic properties of plethystic vertex operators, introduced in J. Phys. A: Math. Theor. 43 405202 (2010), underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group, defined to stabilize tensors of Young symmetry type characterized by a partition of arbitrary shape \pi. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each \pi, the modes generated by the plethystic vertex operators and their suitably constructed duals, satisfy the anticommutation relations of a complex Clifford algebra. The combinatorial manipulations underlying the results involve exchange identities exploiting the Hopf-algebraic structure of certain symmetric function series and their plethysms.

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arXiv2014

Geometric constructions preserve fibrations

Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a 2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i) $\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod}$ and (ii) $\mathfrak{T}_\bullet$ preserves proneness of morphisms in $\mathcal{C}^\downarrow$. Then $\mathfrak{T}_\bullet$ preserves fibrations and opfibrations in $\mathcal{C}$. The proof takes Street's characterization of (e.g.) opfibrations as pseudoalgebras for 2-monads $\mathfrak{L}_B$ on slice categories $\mathcal{C}/B$ and develops it by defining a 2-monad $\mathfrak{L}_\bullet$ on $\mathcal{C}^\downarrow$ that takes change of base into account, and uses known results on the lifting of 2-functors to pseudoalgebras.

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arXiv2014

Smooth coalgebra: testing vector analysis

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to Banach spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

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arXiv2013

Hopf Algebras, Distributive (Laplace) Pairings and Hash Products: A unified approach to tensor product decompositions of group characters

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the Appendix we discuss a relation to formal group laws.

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arXiv2012

The Born rule as structure of spectral bundles (extended abstract)

Topos approaches to quantum foundations are described in a unified way by means of spectral bundles, where the base space is a space of contexts and each fibre is its spectrum. Differences in variance are due to the bundle being a fibration or opfibration. Relative to this structure, the probabilistic predictions of the Born rule in finite dimensional settings are then described as a section of a bundle of valuations. The construction uses in an essential way the geometric nature of the valuation locale monad.

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arXiv2012

Ribbon Hopf algebras from group character rings

We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group GL(n) in the inductive limit, realised as the ring of symmetric functions \Lambda(X) on countably many variables X = {x_1,x_2, ...}. Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of GL(n), the orthogonal and symplectic groups. We also analyse the formal character rings Char-H_\pi\ of algebraic subgroups of GL(n), comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type \pi, which have been introduced in [5] (these include the orthogonal and symplectic groups as special cases). The set of tangle diagrams encoding manipulations of the group and subgroup characters has many elements deriving from products, coproducts, units and counits as well as different types of branching operators. From these elements we assemble for each \pi\ a crossing tangle which satisfies the braid relation and which is nontrivial, in spite of the commutative and co-commutative setting. We identify structural elements and verify the axioms to establish that each Char-H_\pi\ ring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe.

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arXiv2012

On parallelizing the Clifford algebra product for CLIFFORD

We present, as a proof of concept, a way to parallelize the Clifford product in CL_{p,q} for a diagonalized quadratic form as a new procedure `cmulWpar' in the \Clifford package for \Maple(R). The procedure uses a new `Threads' module available under Maple 15 (and later) and a new \Clifford procedure `cmulW' which computes the Clifford product of any two Grassmann monomials in \CL_{p,q} with a help of Walsh functions. We benchmark `cmulWpar' and compare it to two other procedures `cmulNUM' and `cmulRS' from \Clifford. We comment on how to improve `cmulWpar' by taking advantage of multi-core processors and multithreading available in modern processors.

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arXiv2012

Symbolic Computations in Higher Dimensional Clifford Algebras

We present different methods for symbolic computer algebra computations in higher dimensional (\ge9) Clifford algebras using the \Clifford\ and \Bigebra\ packages for \Maple(R). This is achieved using graded tensor decompositions, periodicity theorems and matrix spinor representations over Clifford numbers. We show how to code the graded algebra isomorphisms and the main involutions, and we provide some benchmarks.

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arXiv2012

Some graphical aspects of Frobenius structures

We survey some aspects of Frobenius algebras, Frobenius structures and their relation to finite Hopf algebras using graphical calculus. We focus on the `yanking' moves coming from a closed structure in a rigid monoidal category, the topological move, and the `yanking' coming from the Frobenius bilinear form and its inverse, used e.g. in quantum teleportation. We discus how to interpret the associated information flow. Some care is taken to cover non-symmetric Frobenius algebras and the Nakayama automorphism. We review graphically the Larson-Sweedler-Pareigis theorem showing how integrals of finite Hopf algebras allow to construct Frobenius structures. A few pointers to further literature are given, with a subjective tendency to graphically minded work.

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arXiv2011

Transposition anti-involution in Clifford algebras and invariance groups of scalar products on spinor spaces

We introduce on the abstract level in real Clifford algebras \cl_{p,q} of a non-degenerate quadratic space (V,Q), where Q has signature \epsilon=(p,q), a transposition anti-involution \tp. In a spinor representation, the anti-involution \tp gives transposition, complex Hermitian conjugation or quaternionic Hermitian conjugation when the spinor space \check{S} is viewed as a \cl_{p,q}-left and \check{K}-right module with \check{K} isomorphic to R or R^2, C, or, H or H^2. \tp is a lifting to \cl_{p,q} of an orthogonal involution \tve: V \rightarrow V which depends on the signature of Q. The involution is a symmetric correlatio \tve: V \rightarrow V^{*} \cong V and it allows one to define a reciprocal basis for the dual space (V^{*},Q). The anti-involution \tp acts as reversion on \cl_{p,0} and as conjugation on \cl_{0,q}. Using the concept of a transpose of a linear mapping one can show that if [L_u] is a matrix in the left regular representation of the operator L_u: \cl_{p,q} \rightarrow \cl_{p,q} relative to a Grassmann basis B in \cl_{p,q}, then matrix [L_{\tp(u)}] is the matrix transpose of [L_u]. Of particular importance is the action of \tp on the algebraic spinor space S, generated by a primitive idempotent f, or a sum f+\hat{f} in simple or semisimple algebras. \tp allows us to define a new spinor scalar product S \times S \rightarrow \check{K}, where K=f\cl_{p,q}f and \check{K}=K or K \oplus \hat{K} in the simple or semisimple case. Our scalar product reduces to well known ones in Euclidean and anti-Euclidean signatures. \tp acts as identity, complex conjugation, or quaternionic conjugation on \check{K}. The action of \tp on spinors results in matrix transposition, complex Hermitian conjugation, or quaternionic ermitian conjugation. We classify the automorphism groups of the new product as O(N), U(N), Sp(N), O(N)^2, or Sp(N)^2.

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arXiv2011

On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces

A signature epsilon=(p,q) dependent transposition anti-involution T of real Clifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in [arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on the value of (p-q) mod 8, the map T 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 [Lounesto, Clifford algebras and Spinors 2001]. We provide a full signature (p,q) dependent classification of the invariance groups Gpq_{p,q} of this product for p+q <= 9. The map T is identified as the "star" map known [Passmann, The Algebraic Structure of Group Rings 1985] from the theory of (twisted) group algebras, where the Clifford algebra Cl_{p,q} is seen as a twisted group ring k^t[(Z_2)^n], n=p+q. We discuss and list important subgroups of stabilizer groups Gpq(f)_{p,q} and their transversals in relation to generators of spinor spaces.

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arXiv2010

Plethysms, replicated Schur functions and series, with applications to vertex operators

Specializations of Schur functions are exploited to define and evaluate the Schur functions s_\lambda[\alpha X] and plethysms s_\lambda[\alpha s_\nu(X))] for any \alpha - integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, M_\pi and L_\pi, specified by arbitrary partitions \pi. These are used in turn to define and provide generating functions for formal characters, s_\lambda^{(\pi)}, of certain groups H_\pi, thereby extending known results for orthogonal and symplectic group characters. Each of these formal characters is then given a vertex operator realization, first in terms of the series M=M_{(0)} and various L_\sigma^\perp dual to L_\sigma, and then more explicitly in exponential form. Finally the replicated form of such vertex operators are written down.

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arXiv2010

On the transposition anti-involution in real Clifford algebras I: The transposition map

A particular orthogonal map on a finite dimensional real quadratic vector space (V,Q) with a non-degenerate quadratic form Q of any signature (p,q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra CL(V^*,Q) of linear functionals (multiforms) acting on the universal Clifford algebra CL(V,Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of CL(V,Q). The anti-involution 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 the element matrix in the left regular representation of CL(V,Q). We give also an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [...II: Spabilizer groups of primitive idempotents].

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arXiv2010

On the transposition anti-involution in real Cliffrd algebras II: Stabilizer groups of primitive idempotents

In the first article of this work [... I: The transposition map] we showed that real Clifford algebras CL(V,Q) posses a unique transposition anti-involution \tp. There it was shown that 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 the associated matrix of that element in the left regular representation of the algebra. In this paper we show that, depending on the value of (p-q) mod 8, where \ve=(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 \tp allows us to define a dual spinor space S^\ast, 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.

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arXiv2007

Hopf algebras and characters of classical groups

Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of n-independent universal characters. The corresponding rings, CharGL, CharO and CharSp, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case.

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arXiv2007

The Hopf Algebra Structure of the Character Rings of Classical Groups

The character ring \CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \Sym$ of symmetric functions. Here we study the character rings \CO and \CSp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that \CO and \CSp also admit natural Hopf algebra structures that are isomorphic to that of \CGL, and hence to \Sym. The isomorphisms are determined explicitly, along with the specification of standard bases for \CO and \CSp analogous to those used for \Sym. A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the \CGL case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras \CO and \CSp are not self-dual. The dual Hopf algebras \CO^* and \CSp^* are identified. Finally, the Hopf algebra of the universal rational character ring \CGLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

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arXiv2006

Renormalization : A number theoretical model

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.

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arXiv2005

The Dirichlet Hopf algebra of arithmetics

In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an `unrenormalized' coproduct and an `unrenormalized' pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra. This can be modelled alternatively by employing Rota-Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

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