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.     

Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

In this paper we show how the dynamics of the Schr?dinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, C_1,3, C_3,0{\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}, and C_0,1{\mathcal{C}}_{0,1}. Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.     

Deformations of Algebras Constructed using Quantum Stochastic Calculus

Deformations of associative algebras in which time is the deformation parameter are constructed using quantum stochastic flows in which the usual multiplicativity requirement is replaced by multiplicativity with respect to the deformed multiplication. The theory is restricted by a commutativity requirement on the structure maps of the flow, but examples which can be constructed in this way include the noncommutative torus and the Weyl–Moyal deformation.     

Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type.     

Braided m-Lie Algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End _F M, where M is a Yetter–Drinfeld module over B with dimB < . In particular, generalized classical braided m-Lie algebras sl _{q, f} (GM _G (A), F) and osp _{q, t} (GM _G (A), M, F) of generalized matrix algebra GM _G (A) are constructed and their connection with special generalized matrix Lie superalgebra sl _{s, f} (GM _{Z_2}(A ^s ), F) and orthosymplectic generalized matrix Lie super algebra osp _{s, t} (GM _{Z_2}(A ^s ), M ^s , F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.This revised version was published online in March 2005 with corrections to the cover date.     

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.     

Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras

In this paper, we study the generalized quantum double construction for paired Hopf algebras with particular attention to the case when the generalized quantum double is a Hopf algebra with projection. Applying our theory to a coquasitriangular Hopf algebra (H, σ), we see that H has an associated structure of braided Hopf algebra in the category of Yetter-Drinfeld modules over , where H _σ is a subHopf algebra of H ⁰, the finite dual of H. Specializing to the quantum group H = SL _q (N), we find that H _σ is , so that the duality between these quantum groups is just the evaluation map. Furthermore, we obtain explicit formulas for the braided Hopf algebra structure of SL _q (N) in the category of left Yetter-Drinfeld modules over . The second author held a postdoctoral fellowship at Mount Allison University from 2005 to 2007 and would like to thank Mount Allison for their warm hospitality. Support for the first author’s research and partial support for the postdoctoral position of the second author came from an NSERC Discovery Grant. The second author now holds research support from Grant 434/1.10.2007 of CNCSIS.     

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.     

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.     

Deformations of Vertex Algebras,Quantum Cohomology of Toric Varieties,and Elliptic Genus

 We use previous work on the chiral de Rham complex and Borisov's deformation of a lattice vertex algebra to give a simple linear algebra construction of quantum cohomology of toric varieties. Somewhat unexpectedly, the same technique allows to compute the formal character of the cohomology of the chiral de Rham complex on even dimensional projective spaces. In particular, we prove that the formal character of the space of global sections equals the equivariant signature of the loop space, a well-known example of the Ochanine-Witten elliptic genus. Received: 15 July 2000 / Accepted: 17 August 2002 Published online: 8 January 2003 RID="*" ID="*" Partially supported by an NSF grant Communicated by R. H. Dijkgraaf     

Pauli-Dirac matrix generators of Clifford Algebras

This article presents a Pauli-Dirac matrix approach to Clifford Algebras. It is shown that the algebra C₂ is generated by two Pauli matrices i₂ and i₃; C₃ is generated by the three Pauli matrices ₁, ₂, ₃; C₄ is generated by four Dirac matrices ₀, ₁, ₂, ₃ and C₅ is generated by five Dirac matrices i₀, i₁, i₂, i₃, i₅. The higher dimensional anticommuting matrices which generate arbitrarily high order Clifford algebras are given in closed form. The results obtained with this Clifford algebra approach are compared with the vector product method which was described in a recent article [Found. Phys. 10, 531–553 (1980) by Poole, Farach and Aharonov] and with the Dirac, Rashevskii and Ramakrishnan methods of matrix generation.Supported by the National Science Foundation under Grant ISP-80-11451.     

Braided Quantum Field Theory

We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for n-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and under-crossings. We demonstrate the power of our approach by applying it to φ⁴-theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised. Received: 3 July 1999 / Accepted: 10 November 2000     

Gauge Field Theories on Clifford Algebras

We present a straightforward model of the U(1) gauge equations of Dirac and Maxwell, as well as the U(n) Yang–Mills equations where all fields and gauge transformations take values in a Clifford algebra. When expressed in terms of the Clifford components of the fields, the equations display various gauge symmetries which we intestigate for all Clifford algebras. In particular, for the Pauli algebra, the Dirace CA equations possess the SU(2) × U(1)-symmetry.     

Galilean-Covariant Clifford Algebras in the Phase-Space Representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.     

Quantum Spheres and Projective Spaces as Graph Algebras

 The C ^* -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively. Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation. RID="**" ID="**" Partially supported by the Research Management Committee of the University of Newcastle.     

Formality and Deformations of Universal Enveloping Algebras

We describe enveloping algebras of finite-dimensional Lie algebras which are formal in the sense that their Hochschild complex as a differential graded Lie algebra is quasi-isomorphic to its Hochschild cohomology. For Abelian Lie algebras this is true thanks to the Kontsevich formality theorem. We are using his formality map twisted by the group-like element generated by the linear Poisson structure to simplify the problem, and then study examples. For instance, the universal enveloping algebras of the Lie algebras are formal. We also recover our rigidity results for enveloping algebras from this new angle and present some explicit deformations of linear Poisson structure in low dimensions.     

Quantum Quasi-Shuffle Algebras

We establish some properties of quantum quasi-shuffle algebras. They include the necessary and sufficient condition for the construction of the quantum quasi-shuffle product, the universal property, and the commutativity condition. As an application, we use the quantum quasi-shuffle product to construct a linear basis of T(V), for a special kind of Yang–Baxter algebras (V, m, σ).     

Quantum Implication Algebras

Quantum implication algebras without complementation are formulated with the same axioms for all five quantum implications. Previous formulations of orthoimplication, orthomodular implication, and quasi-implication algebras are analyzed and put in perspective to each other and our results.