Braided Identities, Quantum Groups, and Clifford Algebras

A Lie algebra in a braided category is constructed within the algebra structure of the positive part of the Drinfeld—Jimbo quantum group of type An such that its universal enveloping algebra is a braided Hopf algebra. Similarities with Clifford algebras are discussed.     

De Sitter Quasigroups

The Cayley–Klein parameters for the de Sitter groups SO(4, 1) and SO(3, 2) are introduced, and in an extension of the earlier investigation of quasigroups connected with Clifford groups, quasigroups connected with the SO(4, 1) and SO(3, 2) groups are determined. It is shown that these quasigroups have eight-dimensional, double-valued irreducible cracovian representations. The covariance of a five-dimensional form of the Dirac equation with respect to the quasi-rotations forming quasigroups connected with the groups SO(4, 1) and SO(3, 2) is demonstrated. An analogy is drawn between Weyl's hidden symmetry group and a quasigroup.     

Positive Energy Representations of the Loop Groups of Non-Simply Connected Lie Groups

We classify and construct all irreducible positive energy representations of the loop group of a compact, connectedand simple Lie group and show that they admit an intertwining action of Diff(S ¹). Received: 7 April 1998 / Accepted: 26 April 1999     

A Formula for Pseudocharacters on Almost Connected Groups

An explicit formula for pseudocharacters on almost connected locally compact groups is given.     

Irreducible Locally Bounded Finite-Dimensional Pseudorepresentations of Connected Locally Compact Groups

The irreducible locally bounded finite-dimensional pseudorepresentations of connected locally compact groups are described.     

Continuity Conditions for Finite-Dimensional Locally Bounded Representations of Connected Locally Compact Groups

A criterion for the continuity of the restriction of a finite-dimensional locally bounded representation of a connected locally compact group to the commutator subgroup of the group is given.     

Hyperbolic Conformal Geometry with Clifford Algebra

In this paper, we study hyperbolic conformal geometry following a Clifford algebraic approach. Similar to embedding an affine space into a one-dimensional higher linear space, we embed the hyperboloid model of the hyperbolic n-space in into . The model is convenient for the study of hyperbolic conformal properties. Besides investigating various properties of the model, we also study conformal transformations using their versor representations.     

Idempotents of Clifford Algebras

A classification of idempotents of Clifford algebras C ^p,q is presented. It is shown that using isomorphisms between Clifford algebras C ^p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed.     

Quantum Gates and Quantum Algorithms with Clifford Algebra Technique

We use the Clifford algebra technique (J. Math. Phys. 43:5782, 2002; J. Math. Phys. 44:4817, 2003), that is nilpotents and projectors which are binomials of the Clifford algebra objects γ ^a with the property {γ ^a ,γ ^b }₊=2η ^ab , for representing quantum gates and quantum algorithms needed in quantum computers in a simple and an elegant way. We identify n-qubits with the spinor representations of the group SO(1,3) for a system of n spinors. Representations are expressed in terms of products of projectors and nilpotents; we pay attention also on the nonrelativistic limit. An algorithm for extracting a particular information out of a general superposition of 2 ⁿ qubit states is presented. It reproduces for a particular choice of the initial state the Grover’s algorithm (Proc. 28th Annual ACM Symp. Theory Comput. 212, 1996).     

States on Clifford algebras

We study states on Clifford algebras from the point of view of C*-algebras. A criterium is given under which the odd-point functions vanish. A particular set of states, called quasi-free states is extensively studied and explicit representations are given; as an application we give an approximate calculation of the ground state of a Fermion system.On leave from Matematisk Institut, University of Aarhus, Denmark.Aangesteld navorser van het Belgisch N. F. W. O. On leave from University of Louvain, Belgium.     

Metrics are Clifford algebra involutions

Four-dimensional space-time, all relevant inner products, and some of the groups leaving these inner products invariant are manufactured from more basic algebraic ingredients, all inside the 8-dimensional Pauli algebra : (1) Euclidean 3-spaceE ³, (2) Minkowski 4-spaceM ⁴, (3) complex 4-space ⁴, and all three metrics and all three inner products. The groupsSO(3;) SO(3; 1;) SO (4;) are obtained as images of twofold covering maps of subgroups of or their direct product. A method of embedding in the Clifford algebraC(1;n–1) ofn-dimensional Minkowski space is given for anyn4. Furthermore, all three groups act not only on the relevant vector spaces, but on all ofC(1;n–1), leaving setwise invariant.     

On generalized Clifford groups II

The earlier study of the irreducible representations of the generalized Clifford groups G^m_n in the case where m is a prime number, is now extended to the case where m is any integer. The analysis of class structure and hence the construction of the irreducible representations of G^m_n for a non-prime integer m is found to be more complicated. This investigation also requires the properties of the generalized Clifford algebras C^m_n(I) which are studied in Section 2 of the paper. The case of infinite generalized Clifford group, i.e. G^∞_n involving the infinite- order root of unity as well as the physical relevance of the generalized Clifford groups are briefly dealt with.     

Matrix Exponential via Clifford Algebras

Abstract

Expanded version of a talk presented at the Special Session on ‘Octonions and Clifford Algebras’, 1997 Spring Western Sectional 921st Meeting of the American Mathematical Society, Oregon State University, Corvallis, OR, 19–20 April 1997.

We use isomorphism ? between matrix algebras and simple orthogonal Clifford algebras C (Q) to compute matrix exponential e^A of a real, complex, and quaternionic matrix A. The isomorphic image p = ?(A) in C (Q), where the quadratic form Q has a suitable signature (p, q), is exponentiated modulo a minimal polynomial of p using Clifford exponential. Elements of C (Q) are treated as symbolic multivariate polynomials in Grassmann monomials. Computations in C (Q) are performed with a Maple package ‘CLIFFORD’. Three examples of matrix exponentiation are given.     

Linear operators on Clifford algebras

The linear operators from a finite-dimensional Clifford algebra to itself are computed in terms of suitable two-sided multiplications and operations on the center.     

Linear operators in Clifford algebras

We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M _{eI, eJ} defined by xe _I · x · e _J , where the e _I are the generators of the Clifford algebra and I is a multi-index (3, 7).In particular, it is shown that the family (M _{eI, eJ} ) is exactly a basis in the even case.     

Clifford hopf-gebra and bi-universal hopf-gebra

We consider a pair of independent scalar products, one scalar product on vectors, and another independent scalar product on dual space of co-vectors. The Clifford co-product of multivectors is calculated from the dual Clifford algebra. With respect to this co-product unit is not group-like and vectors are not primitive. The Clifford product and the Clifford co-product fits to the bi-gebra with respect to the family of the (pre)-braids. The Clifford bi-gebra is in a braided category iff at least one of these scalar products vanish. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. This paper is in final form and no version of it will be submitted for publication elsewhere; q-alg/9709016.     

Signature Change and Clifford Algebras

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the Minkowskian Dirac equation over the Euclidean spacetime and to define a new duality operation in terms of which one can find self-dual and anti-self-dual solutions of gauge fields over Minkowski spacetime analogous to the ones over Euclidean spacetime and without needing to complexify the original real algebra.     

Clifford algebras and Hestenes spinors

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl _1,3 M₂ (H). Hestenes invented first in 1966 hisideal spinors and later 1967/75 he recognized the importance of hisoperator spinors Cl _1,3 ⁺ M₂ (C).This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC Cl _1,3 andnot inC M₄ (R)=M₄ (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor.Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ₀=0, ₀₀₁₂₃=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation.Projection operators P_±, _± are resettled among their new relatives in End(Cl _1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl _1,3 to the (graded) opposite algebra Cl _3,1 without resorting to complex numbers, that is, not using a replacement i.     

Countably Solvable Connected Pro-Lie Groups Are <Emphasis Type="Italic">u</Emphasis>-Amenable

In contrast to some well-known discrete groups, countably solvable connected pro-Lie groups are u-amenable in the sense of de la Harpe’s 1973 paper.