48,250 research outputs found
On projective representations for compact quantum groups
We study actions of compact quantum groups on type I factors, which may be
interpreted as projective representations of compact quantum groups. We
generalize to this setting some of Woronowicz' results concerning Peter-Weyl
theory for compact quantum groups. The main new phenomenon is that for general
compact quantum groups (more precisely, those which are not of Kac type), not
all irreducible projective representations have to be finite-dimensional. As
applications, we consider the theory of projective representations for the
compact quantum groups associated to group von Neumann algebras of discrete
groups, and consider a certain non-trivial projective representation for
quantum SU(2).Comment: 43 page
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices behave almost as well as matrices with commutative elements.
Namely theorems of linear algebra (e.g., a natural definition of the
determinant, the Cayley-Hamilton theorem, the Newton identities and so on and
so forth) holds true for them.
On the other hand, we remark that such matrices are somewhat ubiquitous in
the theory of quantum integrability. For instance, Manin matrices (and their
q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and
the so--called Cartier-Foata matrices. Also, they enter Talalaev's
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show
that theorems of linear algebra, after being established for such matrices,
have various applications to quantum integrable systems and Lie algebras, e.g
in the construction of new generators in (and, in general,
in the construction of quantum conservation laws), in the
Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also
discuss applications to the separation of variables problem, new Capelli
identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
Heat Kernel Asymptotics on Homogeneous Bundles
We consider Laplacians acting on sections of homogeneous vector bundles over
symmetric spaces. By using an integral representation of the heat semi-group we
find a formal solution for the heat kernel diagonal that gives a generating
function for the whole sequence of heat invariants. We argue that the obtained
formal solution correctly reproduces the exact heat kernel diagonal after a
suitable regularization and analytical continuation.Comment: 29 pages, Proceedings of the 2007 Midwest Geometry Conference in
Honor of Thomas P. Branso
Magnetic fields in noncommutative quantum mechanics
We discuss various descriptions of a quantum particle on noncommutative space
in a (possibly non-constant) magnetic field. We have tried to present the basic
facts in a unified and synthetic manner, and to clarify the relationship
between various approaches and results that are scattered in the literature.Comment: Dedicated to the memory of Julius Wess. Work presented by F. Gieres
at the conference `Non-commutative Geometry and Physics' (Orsay, April 2007
Canonical approach to 2D WZNW model, non-abelian bosonization and anomalies
The gauged WZNW model has been derived as an effective action, whose Poisson
bracket algebra of the constraints is isomorphic to the commutator algebra of
operators in quantized fermionic theory. As a consequence, the hamiltonian as
well as usual lagrangian non-abelian bosonization rules have been obtained, for
the chiral currents and for the chiral densities. The expression for the
anomaly has been obtained as a function of the Schwinger term, using canonical
methods.Comment: RevTex, 23 page
Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)
In [CPR2], we presented a K-theoretic approach to finding invariants of
algebras with no non-trivial traces. This paper presents a new example that is
more typical of the generic situation. This is the case of an algebra that
admits only non-faithful traces, namely SU_q(2), and also KMS states. Our main
results are index theorems (which calculate spectral flow), one using ordinary
cyclic cohomology and the other using twisted cyclic cohomology, where the
twisting comes from the generator of the modular group of the Haar state. In
contrast to the Cuntz algebras studied in [CPR2], the computations are
considerably more complex and interesting, because there are nontrivial `eta'
contributions to this index.Comment: 25 pages, 1 figur
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