Differential geometry of the quantum supergroupGL q (1/1)

We construct a right-invariant differential calculus on the quantum supergroupGL _q (1/1) and we show that the quantum Lie algebra generators satisfy the undeformed Lie superalgebra. The deformation becomes apparent when one studies the comultiplication for these generators. We bring the algebra into the standard Drinfeld-Jimbo form by performing a suitable change of variables, and we check the consistency of the map with the induced comultiplication.     

Non-Commutative Batalin-Vilkovisky Algebras,Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Δ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the “big bracket” of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.     

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

Lie Algebras of Derivations and Resolvent Algebras

This paper analyzes the action δ of a Lie algebra X by derivations on a C*–algebra ${\mathcal{A}}$ . This action satisfies an “almost inner” property which ensures affiliation of the generators of the derivations δ with ${\mathcal{A}}$ , and is expressed in terms of corresponding pseudo–resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*–algebra ${\mathcal{A}}$ , it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo–resolvents. If the Lie action δ is ergodic, i.e. the only elements of ${\mathcal{A}}$ on which all the derivations in δ _X vanish are multiples of the identity, then this extension is given by a (non–degenerate) symplectic form σ on X. Moreover, the algebra generated by the pseudo–resolvents coincides with the resolvent algebra based on the symplectic space (X, σ). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*–algebras.     

Group manifold analysis of the structure of relativistic quantum dynamics

We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincaré group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.     

A bicovariant differential algebra of a quantum group

A bicovariant differential algebra of four basic objects (coordinate functions, differential 1-forms, Lie derivatives and inner derivations) within a differential calculus on a quantum group is shown to be produced by a direct application of the cross-product construction to the Woronowicz differential complex, whose Hopf algebra properties account for the bicovariance of the algebra. A correspondence with classical differential calculus, including Cartan identity, and some other useful relations are considered. An explicit construction of a bicovariant differential algebra on GL_q(N) is given and its (co)module properties are discussed.     

Lie-admissible structures on Witt type algebras

In this paper we study Lie-admissible structures on Witt type algebras. Witt type algebras are Γ$\Gamma$-graded Lie algebras (where Γ$\Gamma$ is an abelian group) which generalize the Witt algebra. We give all third power-associative and flexible Lie-admissible structures on these algebras. In particular we generalize some results on the Witt algebra. After describing the second scalar cohomology group of Witt type algebras, we investigate third power-associative and flexible Lie-admissible structures on the central extension of some Witt type algebras. Finally we study a left-symmetric structure induced by a symplectic form for some Witt type algebras.     

Monodromy properties of energy momentum tensor on general algebraic curves

A new approach to analyze the properties of the energy momentum tensor T (z) of conformal field theories on generic Riemann surfaces (RS) is proposed. T (z) is decomposed into N components with different monodromy properties, where N is the number of branches in the realization of RS as branch covering over the complex sphere. This decomposition gives rise to new infinite dimensional Lie algebra which can be viewed as a generalization of Virasoro algebra containing information about the global properties of the underlying RS. In the simplest case of hyperelliptic curves the structure of the algebra is calculated in two ways and its central extension is explicitly given. The algebra possess an interesting symmetry with a clear interpretation in the framework of the radial quantization of CFTs with multivalued fields on the complex sphere.     

On a microscopic representation of space-time

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low-energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and ??reduce?? the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) ?? usp(4) ?? su(2) × u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space H ₅ with SO(5, 1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into H ₂ ?? H ₃ with SO(2, 1) and SO(3, 1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra H and subsequent coordinate restrictions.     

Differential algebra of quantum fermions

Lie coalgebra equips an exterior algebra (algebra of fermions) with a structure of a differential algebra. In similar way we equip an algebra of quantum fermions (quantized exterior algebra) with a structure of a differential algebra. This leads to a notion of a variety of Lie coalgebras for a Hecke braid. This approach is different from that of Gurevich (1988 and 1993), Woronowicz (1989) and of Majid (1993).     

Relativistic internal time operator

We construct a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincaré group. The Lie algebra generated by the time operator and the generators of the Poincaré group turns out to be an infinitedimensional extension of the Poincaré algebra. The internal time operator generates two new entities, namely the velocity operator and the internal position operator. The transformation properties of the internal time and position operator under Lorentz boosts are different from what one would expect from relativity theory. This difference reflects the fact that the time concept associated with the internal time operator is radically different from the time coordinate of Minkowski space, due to the nonlocality of the time operator. The spectral projections of the time operator allow us to construct incoming subspaces for the wave equation without invoking Huygens' principle, as in two and one spatial dimensions where Huygens' principle does not hold.     

Exterior calculus with d 3=0 on two-dimensional quantum plane

In this paper, we construct the algebra of differential forms with exterior differential satisfying d ³=0 on the two-dimensional quantum plane assuming that the homomorphism defining first-order differential calculus is linear in variables. Assuming d ²≠0, we introduce the second-order differentials d ² x ⁱ. The commutation relations between the generators x ⁱ, dx ⁱ, and d ² x ⁱ of the algebra of differential forms, among dx ⁱ, and among d ² x ⁱ, as well as between noncommutative derivatives with generators, are found. The consistency conditions are described.     

A symplectic context for level dynamics

We consider a mathematical context which was suggested by quantum mechanical considerations of level dynamics. Although the situation is a general one, we restrict our attention to certain examples of physical relevance where explicit calculations are possible. Cases where M is the cotangent space of some Lie group or Lie algebra Q of operators on a finite-dimensional vector space are of particular interest.     

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q ². In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ ₊=P _? QP ₊, whereP _±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q ₊) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q ₊)=n?1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.     

Duality on the quantum space(3) with two parameters

In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.     

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.     

On the associativity anomaly in open string field theory

The associativity anomaly in the star algebra of the open bosonic string is demonstrated by a simple oscillator calculation. In an associative algebra, formal arguments require that the “half string” BRST operator Q_L be nilpotent; however, we show that associativity is actually violated by computing Q_L² in an explicit operator representation of the star algebra.