1,720,973 research outputs found
Centres, trace functors, and cyclic cohomology
Full text linkWe study the biclosedness of the monoidal categories of modules and comodules
over a (left or right) Hopf algebroid, along with the bimodule category centres
of the respective opposite categories and a corresponding categorical
equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd
modules, respectively. This is directly connected to the existence of a trace
functor on the monoidal categories of modules and comodules in question, which
in turn allows to recover (or define) cyclic operators enabling cyclic
cohomology.Comment: 41 pages; v2: added a subsection which better elucidates how trace
functors emerge in biclosed categories; various minor improvements. To appear
in Comm. Contemp. Mat
A noncommutative calculus on the cyclic dual of Ext
No full textWe show that if the cochain complex computing Ext groups (in the category
of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative
Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on
Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces
computing the two classical derived functors lead to complexes that compute the more exotic
ones, giving a cyclic opposite module over an operad with multiplication that induce
operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential,
along with higher homotopy operators defining a noncommutative Cartan calculus up
to homotopy. In particular, this allows to recover the classical Cartan calculus from differential
geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without
any finiteness conditions or the use of topological tensor products
Gerstenhaber and Batalin-Vilkovisky Structures on Modules over Operads
Full text linkIn this article, we show under what additional ingredients a comp (or opposite) module
over an operad with multiplication can be given the structure of a cyclic k-module and
how the underlying simplicial homology gives rise to a Batalin–Vilkovisky module over
the cohomology of the operad. In particular, one obtains a generalized Lie derivative and
a generalized (cyclic) cap product that obey a Cartan–Rinehart homotopy formula, and
hence yield the structure of a noncommutative differential calculus in the sense of Nest,
Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild
theory of associative algebras, for Poisson structures, but above all the calculus for
general left Hopf algebroids with respect to general coefficients (in which the classical
calculus of vector fields and differential forms is contained)
Duality and products in algebraic (co)homology theories
No full textThe origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a ×A-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie–Rinehart (co)homology
Cyclic Gerstenhaber-Schack cohomology
Full text linkWe show that the diagonal complex computing the Gerstenhaber-Schack cohomology
of a bialgebra (that is, the cohomology theory governing bialgebra deformations)
can be given the structure of an operad with multiplication if the bialgebra is a (not
necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is
involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of
any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkoviski ̆ı algebra structure;
in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber
bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case
the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to
be zero in cohomology and hence the interesting structure is not given by this e2-algebra
structure but rather by the resulting e3-algebra structure, which is expressed in terms of
the cup product and B
Batalin-Vilkovisky structures on Ext and Tor
No full textThis article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and etale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on Ext_U(A,A) are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan-Rinehart homotopy formula whose essence is that Tor^U(M,A) becomes for suitable right U-modules M a Batalin-Vilkovisky module over Ext_U(A,A), or in the words of Nest, Tamarkin, Tsygan and others, that Ext_U(A,A) and Tor^U(M,A) form a differential calculus. As an illustration, we show how the well-known operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra to twisted Calabi-Yau algebras
Cyclic structures in algebraic cohomology theories
No full textThis note discusses the cyclic cohomology of a left Hopf algebroid (-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd modules
Phase space reduction of star products on cotangent bundles
No full textIn this paper we construct star products on Marsden-Weinstein reduced
spaces in case both the original phase space and the reduced phase space are (symplectomorphic
to) cotangent bundles. Under the assumption that the original cotangent
bundle T∗Q carries a symplectic structure of form ωB0 = ω0 +π∗B0 with B0
a closed two-form on Q, is equipped by the cotangent lift of a proper and free Lie
group action on Q and by an invariant star product that admits a G-equivariant
quantum momentum map, we show that the reduced phase space inherits from
T∗Q a star product. Moreover, we provide a concrete description of the resulting
star product in terms of the initial star product on T∗Q and prove that our
reduction scheme is independent of the characteristic class of the initial star product.
Unlike other existing reduction schemes we are thus able to reduce not only
strongly invariant star products. Furthermore in this article, we establish a relation
between the characteristic class of the original star product and the characteristic
class of the reduced star product and provide a classification up to G-equivalence
of those star products on (T∗Q, ωB0 ), which are invariant with respect to a lifted
Lie group action. Finally, we investigate the question under which circumstances
‘quantization commutes with reduction’ and show that in our examples non-trivial
restrictions arise
Higher brackets on cyclic and negative cyclic (co)homology
Full text linkThe purpose of this article is to embed the string topology bracket developed by Chas–
Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket
found by de Thanhoffer de Völcsey–Van den Bergh on negative cyclic homology groups
into the global picture of a noncommutative differential (or Cartan) calculus up to
homotopy on the (co)cyclic bicomplex in general, in case a certain Poincaré duality is
given. For negative cyclic cohomology, this in particular leads to a Batalin–Vilkoviski ̆ı
(BV) algebra structure on the underlying Hochschild cohomology. In the special case
in which this BV bracket vanishes, one obtains an e3-algebra structure on Hochschild
cohomology. The results are given in the general and unifying setting of (opposite) cyclic
modules over (cyclic) operads
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