A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 365-427 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Positive Cone ◽  
Point Of View ◽  
Order Structure ◽  
Topological Spaces ◽  
Projective Modules ◽  
Finitely Generated ◽  
C Algebras ◽  
Isomorphism Classes

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

scholarly journals CATEGORIFICATION OF QUANTUM GENERALIZED KAC–MOODY ALGEBRAS AND CRYSTAL BASES

2012 ◽  
Vol 23 (11) ◽  
pp. 1250116 ◽  
Author(s):  
SEOK-JIN KANG ◽  
SE-JIN OH ◽  
EUIYONG PARK
Keyword(s):  
Crystal Structures ◽  
Grothendieck Group ◽  
Integral Form ◽  
Cartan Matrix ◽  
Algebra Homomorphism ◽  
Finitely Generated ◽  
Moody Algebra ◽  
Graded Modules ◽  
Isomorphism Classes ◽  
Highest Weight

We construct and investigate the structure of the Khovanov-Lauda–Rouquier algebras R and their cyclotomic quotients Rλ which give a categorification of quantum generalized Kac–Moody algebras. Let U𝔸(𝔤) be the integral form of the quantum generalized Kac–Moody algebra associated with a Borcherds–Cartan matrix A = (aij)i, j ∈ I and let K0(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism [Formula: see text] and that Φ is an isomorphism if aii ≠ 0 for all i ∈ I. Let B(∞) and B(λ) be the crystals of [Formula: see text] and V(λ), respectively, where V(λ) is the irreducible highest weight Uq(𝔤)-module. We denote by 𝔅(∞) and 𝔅(λ) the isomorphism classes of irreducible graded modules over R and Rλ, respectively. If aii ≠ 0 for all i ∈ I, we define the Uq(𝔤)-crystal structures on 𝔅(∞) and 𝔅(λ), and show that there exist crystal isomorphisms 𝔅(∞) ≃ B(∞) and 𝔅(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac–Moody algebras.

PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS

2003 ◽  
Vol 02 (04) ◽  
pp. 435-449 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
FRANZ HALTER-KOCH
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Isomorphism Classes

We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

Grothendieck groups for categories of complexes

2008 ◽  
Vol 3 (1) ◽  
pp. 165-203 ◽  
Author(s):  
Hans-Bjørn Foxby ◽  
Esben Bistrup Halvorsen
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Full Subcategory ◽  
Grothendieck Group ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Projective Modules ◽  
Finitely Generated ◽  
Grothendieck Groups ◽  
Finitely Generated Modules

AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

scholarly journals A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

2014 ◽  
Vol 14 (02) ◽  
pp. 1550016 ◽  
Author(s):  
N. R. Baeth ◽  
A. Geroldinger ◽  
D. J. Grynkiewicz ◽  
D. Smertnig
Keyword(s):  
Combinatorial Problems ◽  
Point Of View ◽  
Unique Element ◽  
Theoretical Point ◽  
Finitely Generated ◽  
Prime Rings ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Krull Monoid

Let R be a ring and let [Formula: see text] be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let [Formula: see text] denote a set of representatives of isomorphism classes in [Formula: see text] and, for any module M in [Formula: see text], let [M] denote the unique element in [Formula: see text] isomorphic to M. Then [Formula: see text] is a reduced commutative semigroup with operation defined by [M] + [N] = [M ⊕ N], and this semigroup carries all information about direct-sum decompositions of modules in [Formula: see text]. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if End R(M) is semilocal for all [Formula: see text], then [Formula: see text] is a Krull monoid. Suppose that the monoid [Formula: see text] is Krull with a finitely generated class group (for example, when [Formula: see text] is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case, we study the arithmetic of [Formula: see text] using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid [Formula: see text] for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

K-theory of real algebraic surfaces and threefolds

1989 ◽  
Vol 106 (3) ◽  
pp. 471-480 ◽  
Author(s):  
J. Bochnak ◽  
W. Kucharz
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Algebraic Surfaces ◽  
Real Algebraic Variety ◽  
Projective Modules ◽  
Algebraic Subset ◽  
Witt Group ◽  
Regular Functions ◽  
Real Algebraic Surfaces ◽  
Continuous Complex

LetXbe an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Letdenote the ring of regular functions onX([6], chapter 3). (IfXis an algebraic subset of ℝnthenis comprised of all functions of the formf/g, whereg, f: X→ ℝ are polynomial functions withg−1(O) = Ø.) In this paper, assuming thatXis compact, non-singular, and that dimX≤ 3, we compute the Grothendieck groupof projective modules over(cf. Section 1), and the Grothendieck groupand the Witt groupof symplectic spaces over(cf. Section 2), in terms of the algebraic cohomology groupsandgenerated by the cohomology classes associated with the algebraic subvarieties ofX. We also relate the groupto the Grothendieck groupKO(X) of continuous real vector bundles overX, and the groupsandto the Grothendieck groupK(X)of continuous complex vector bundles overX.

scholarly journals K-theory and phase transitions at high energies

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
T. V. Obikhod
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Soliton Solutions ◽  
Complex Vector ◽  
Supergravity Theory ◽  
Isomorphism Classes ◽  
High Energies ◽  
Complete Set ◽  
K Theory ◽  
Calabi Yau Manifolds

The duality between E8xE8 heteritic string on manifold K3xT2 and Type IIA string compactified on a Calabi-Yau manifold induces a correspondence between vector bundles on K3xT2 and Calabi-Yau manifolds. Vector bundles over compact base space K3xT2 form the set of isomorphism classes, which is a semi-ring under the operation of Whitney sum and tensor product. The construction of semi-ring V ect X of isomorphism classes of complex vector bundles over X leads to the ring KX = K(V ect X), called Grothendieck group. As K3 has no isometries and no non-trivial one-cycles, so vector bundle winding modes arise from the T2 compactification. Since we have focused on supergravity in d = 11, there exist solutions in d = 10 for which space-time is Minkowski space and extra dimensions are K3xT2. The complete set of soliton solutions of supergravity theory is characterized by RR charges, identified by K-theory. Toric presentation of Calabi-Yau through Batyrev's toric approximation enables us to connect transitions between Calabi-Yau manifolds, classified by enhanced symmetry group, with K-theory classification.

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals Adelic descent theory

2017 ◽  
Vol 153 (8) ◽  
pp. 1706-1746
Author(s):  
Michael Groechenig
Keyword(s):  
Vector Bundles ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Reductive Algebraic Group ◽  
Perfect Complexes ◽  
Descent Theory ◽  
Closed Fields ◽  
Ring Of Continuous Functions ◽  
Reconstruction Theorem ◽  
Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

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