zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Vector bundles on an elliptic curve and Sklyanin algebras. (English) Zbl 0916.16014
Feigin, B. (ed.) et al., Topics in quantum groups and finite-type invariants. Mathematics at the Independent University of Moscow. Providence, RI: American Mathematical Society. Transl. Math. Monogr. 185(38), 65-84 (1998).
Let $\Cal E$ be an elliptic curve, $\tau\in{\Cal E}$, $0<k<n$ integers with $\gcd(k,n)=1$. To this datum, the authors attached [in Preprint Inst. Theor. Phys., Kiev (1989); see also Funct. Anal. Appl. 23, No. 3, 207-214 (1989); translation from Funkts. Anal. Prilozh. 23, No. 3, 45-54 (1989; Zbl 0687.17001)] an associative algebra $Q_{n,k}(E,\tau)$ generalizing previous work of Sklyanin. In the classical limit “$\tau\to 0$” the algebra $Q_{n,k}(E,\tau)$ becomes abelian and determines a Hamiltonian structure on $\bbfC\bbfP^{n-1}$. One of the main results of the present paper is the determination of the symplectic leaves of this structure in terms of moduli spaces of bundles on $\Cal E$. Given an indecomposable bundle $\xi_{n,k}$ of rank $n$ and degree $k>0$, the moduli space of vector bundles $Y$ with a sub-bundle $(\nu,\rho)\simeq \xi_{0,1}$ and quotient $Y/(\nu,\rho)\simeq\xi_{n,k}$ is isomorphic to $\bbfP(\text{Ext}(\xi_{0,1},\xi_{n,k}))$. The decomposition of this moduli space as a union of strata, where each stratum corresponds to a type of $k+1$-dimensional bundles, coincides with the decomposition into the union of symplectic leaves. The authors consider also the more general situation of moduli spaces of $P$-bundles on $\Cal E$, where $P$ is a parabolic subgroup of a Kac-Moody group $G$, with Hamiltonian structure coming form the standard Lie bialgebra structure on $\text{Lie }G$. They address the question of the combinatorial structure of the stratification of $\bbfP(\text{Ext}(A,B))$, where $A$ and $B$ are bundles on $\Cal E$. Then some associative algebras are introduced, generalizing $Q_{n,k}(E,\tau)$; they allow to quantize the above mentioned Hamiltonian structures in the case when $P$ is a Borel subgroup of $G$. For the entire collection see [Zbl 0892.00020].

16S80Deformation theory of associative ring and algebras
14H52Elliptic curves
14H60Vector bundles on curves and their moduli
16W30Hopf algebras (associative rings and algebras) (MSC2000)
17B37Quantum groups and related deformations
17A45Quadratic algebras (but not quadratic Jordan algebras)