Central Elements in the Universal Enveloping Algebras for the Split Realization of the Orthogonal Lie Algebras

We construct central elements in the universal enveloping algebra using column-determinants for the split realization of the orthogonal Lie algebra. Our central elements are quite new and simple, though they are closely related to what Howe and Umeda gave for the orthogonal Lie algebra under the different realization as the alternating matrices.     

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 Clifford Algebras as Quantum Deformations

We present a general algebraic framework for the study of quantum/braided Clifford algebras. We allow that the quadratic form g on the base vector space takes values from a noncommutative algebra . Clifford algebra is understood as a Chevalley—Kähler deformation of the braided exterior algebra built from V, , and the initial braid operator : . The new product is canonically associated to g, , and , and it is constructed by applying Rota's and Stein Cliffordization.     

Universal Construction of Algebras

We present a direct construction of the abstract generators for q-deformed W_N{\cal W}_N algebras. New quantum algebraic structures of W_q,p{\cal W}_{q,p} type are thus obtained. This procedure hinges upon a twisted trace formula for the elliptic algebra \elp\elp generalizing the previously known formulae for quantum groups. It represents the q-deformation of the construction of W_N{\cal W}_N algebras from Lie algebras.     

Coisotropic Deformations of Associative Algebras and Dispersionless Integrable Hierarchies

The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and Hirota’s tau function, and shows that the dispersionless Hirota bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham equations of genus zero due to Krichever.     

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.     

Formal Deformations,Contractions and Moduli Spaces of Lie Algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute the jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions. The research of the authors was partially supported by grants from the Mathematisches Forschungsinstitut Oberwolfach, OTKA T043641, T043034 and the University of Wisconsin-Eau Claire.     

Universal Differential Calculus on Ternary Algebras

General concept of ternary algebras is introduced in this article, along with several examples of its realization. Universal envelope of such algebras is defined, as well as the concept of tri-modules over ternary algebras. The universal differential calculus on these structures is then defined and its basic properties investigated.     

Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition

Quantum bialgebras derivable from U _q (sl ₂) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition. Dedicated to the memory of our colleague Leonid L. Vaksman (1951–2007) On leave of absence from: TheoryGroup, Nuclear Physics Laboratory,V.N.Karazin Kharkov National University, Svoboda Sq. 4, Kharkov 61077, Ukraine. E-mail: sduplij@gmail.com;     

Spinor and Oscillator Representations of Quantum Enveloping Algebras of Type Bn,Cn and Dn

With the help of the q-deformed bosonic and fermionic oscillation operators, which can be conetructed from the ordinary ones, the quantum enveloping algebras of the classical Lie algebras B, C and D are written down explicitly. Under these representations the highest roots are given.     

Global Geometric Deformations of the Virasoro Algebra,Current and Affine Algebras by Krichever–Novikov Type Algebras

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever–Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac–Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess–Zumino–Witten–Novikov models appearing in the quantization of Conformal Field Theory.     

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     

Ergodic Actions of Universal Quantum Groups on Operator Algebras

We construct ergodic actions of compact quantum groups on C^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1) an ergodic action of the compact quantum A_u(Q) on the type III_u Powers factor R_u for an appropriate positive Q ] GL(2, Â); (2) an ergodic action of the compact quantum group A_u(n) on the hyperfinite II₁ factor R; (3) an ergodic action of the compact quantum group A_u(Q) on the Cuntz algebra _boxclose_boxclose{\cal O}_n for each positive matrix Q ] GL(n, ³); (4) ergodic actions of compact quantum groups on their homogeneous spaces, as well as an example of a non-homogeneous classical space that admits an ergodic action of a compact quantum group.     

Poisson Action and Formality

Using a formality on a Poisson manifold, we construct a star product and for each Poisson vector field a derivation of this star product. Starting with a Poisson action of a Lie group, we are able under a natural cohomological assumption to define a representation of its Lie algebra in the space of derivations of the star product. Finally, we use these results to define some generically tangential star products on duals of Lie algebra as in [1] but in a more realistic context. This work was supported by the CMCU contract 00 F 15 02.     

Two-Parameter Deformations of SU(2) and SU (1,l) Algebras

The algebras SU(2) and SU(1,1) are promoted to two-parameter quantum universal enveloping algebras (QUEA) by a doubleparameter deformation in this paper. The discrete unitary irreducible representations and their deformed coherent states are studied. The deformed generators are given by a Jordan-Schwinger realization and a Bargmann-Fock representation. It is also interesting that the two-parameter deformed coherent states are found to relate to the oneparameter deformed ones by a simple scaling transformation and this can be used to derive the completeness relation of the former.     

Formality Conjectures and Deformation

Let M be a Poisson manifold. Kontsevich proved that star products exist on M and he gave a classification. To relate his classification with other classifications, one could try to extend the Connes–Flato–Sternheimer invariant to a general Poisson manifold. We show how generalization of this invariant is related to the formality conjecture for chains. Finally, we show how to prove those conjectures step by step. Our approach, different from Tamarkin's, will give explicit formulas but doesn't yet solve the general conjecture.     

Cyclic Formality and Index Theorems

The Letter announces the following results (the proofs will appear elsewhere). An operad acting on Hochschild chains and cochains of an associative algebra is constructed. This operad is formal. In the case when this algebra is the algebra of smooth function on a smooth manifold, the action of this operad on the corresponding Hochschild chains and cochains is formal. The induced map on the (periodic) cyclic homology is given by the formula involving the A-genus. The index theorem for degenerate Poisson structures follows from the latter fact.     

Chaotic Expansions of Elements of the Universal Enveloping Superalgebra Associated with a Z2-graded Quantum Stochastic Calculus

Given a polynomial function f of classical stochastic integrator processes whose differentials satisfy a closed Ito multiplication table, we can express the stochastic derivative of f as

Formality Theorem for Quantizations of Lie Bialgebras

Using the theory of props we prove a formality theorem associated with universal quantizations of Lie bialgebras.