Relevant bibliographies by topics / Noncommutative algebras

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Noncommutative algebras'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 August 2024

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Journal articles on the topic "Noncommutative algebras"

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Arutyunov, A. A. "Derivation Algebra in Noncommutative Group Algebras." Proceedings of the Steklov Institute of Mathematics 308, no. 1 (January 2020): 22–34. http://dx.doi.org/10.1134/s0081543820010022.

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Zhou, Chaoyuan. "Acyclic Complexes and Graded Algebras." Mathematics 11, no. 14 (July 19, 2023): 3167. http://dx.doi.org/10.3390/math11143167.

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We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative N-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra.
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Abel, Mati, and Krzysztof Jarosz. "Noncommutative uniform algebras." Studia Mathematica 162, no. 3 (2004): 213–18. http://dx.doi.org/10.4064/sm162-3-2.

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Xu, Ping. "Noncommutative Poisson Algebras." American Journal of Mathematics 116, no. 1 (February 1994): 101. http://dx.doi.org/10.2307/2374983.

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Roh, Jaiok, and Ick-Soon Chang. "Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras." Abstract and Applied Analysis 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/594075.

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We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras.
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Ercolessi, Elisa, Giovanni Landi, and Paulo Teotonio-Sobrinho. "Noncommutative Lattices and the Algebras of Their Continuous Functions." Reviews in Mathematical Physics 10, no. 04 (May 1998): 439–66. http://dx.doi.org/10.1142/s0129055x98000148.

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Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces of noncommutative C*-algebras. These noncommutative algebras play the same rôle as the algebra of continuous functions [Formula: see text] on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C*-algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.
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Ferreira, Vitor O., Jairo Z. Gonçalves, and Javier Sánchez. "Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras." International Journal of Algebra and Computation 25, no. 06 (September 2015): 1075–106. http://dx.doi.org/10.1142/s0218196715500319.

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For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring 𝔇(L) constructed by Lichtman. If U(L) is an Ore domain, 𝔇(L) coincides with its ring of fractions. It is well known that the principal involution of L, x ↦ -x, can be extended to an involution of U(L), and Cimpric proved that this involution can be extended to one on 𝔇(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that 𝔇(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.
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Liang, Shi-Dong, and Matthew J. Lake. "An Introduction to Noncommutative Physics." Physics 5, no. 2 (April 18, 2023): 436–60. http://dx.doi.org/10.3390/physics5020031.

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Noncommutativity in physics has a long history, tracing back to classical mechanics. In recent years, many new developments in theoretical physics, and in practical applications rely on different techniques of noncommutative algebras. In this review, we introduce the basic concepts and techniques of noncommutative physics in a range of areas, including classical physics, condensed matter systems, statistical mechanics, and quantum mechanics, and we present some important examples of noncommutative algebras, including the classical Poisson brackets, the Heisenberg algebra, Lie and Clifford algebras, the Dirac algebra, and the Snyder and Nambu algebras. Potential applications of noncommutative structures in high-energy physics and gravitational theory are also discussed. In particular, we review the formalism of noncommutative quantum mechanics based on the Seiberg–Witten map and propose a parameterization scheme to associate the noncommutative parameters with the Planck length and the cosmological constant. We show that noncommutativity gives rise to an effective gauge field, in the Schrödinger and Pauli equations. This term breaks translation and rotational symmetries in the noncommutative phase space, generating intrinsic quantum fluctuations of the velocity and acceleration, even for free particles. This review is intended as an introduction to noncommutative phenomenology for physicists, as well as a basic introduction to the mathematical formalisms underlying these effects.
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Mahanta, Snigdhayan. "Noncommutative stable homotopy and stable infinity categories." Journal of Topology and Analysis 07, no. 01 (December 2, 2014): 135–65. http://dx.doi.org/10.1142/s1793525315500077.

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The noncommutative stable homotopy category NSH is a triangulated category that is the universal receptacle for triangulated homology theories on separable C*-algebras. We show that the triangulated category NSH is topological as defined by Schwede using the formalism of (stable) infinity categories. More precisely, we construct a stable presentable infinity category of noncommutative spectra and show that NSHop sits inside its homotopy category as a full triangulated subcategory, from which the above result can be deduced. We also introduce a presentable infinity category of noncommutative pointed spaces that subsumes C*-algebras and define the noncommutative stable (co)homotopy groups of such noncommutative spaces generalizing earlier definitions for separable C*-algebras. The triangulated homotopy category of noncommutative spectra admits (co)products and satisfies Brown representability. These properties enable us to analyze neatly the behavior of the noncommutative stable (co)homotopy groups with respect to certain (co)limits. Along the way we obtain infinity categorical models for some well-known bivariant homology theories like KK-theory, E-theory, and connective E-theory via suitable (co)localizations. The stable infinity category of noncommutative spectra can also be used to produce new examples of generalized (co)homology theories for noncommutative spaces.
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LETZTER, EDWARD S. "NONCOMMUTATIVE IMAGES OF COMMUTATIVE SPECTRA." Journal of Algebra and Its Applications 07, no. 05 (October 2008): 535–52. http://dx.doi.org/10.1142/s0219498808002941.

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We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially "sewn together" to form Spec R. In particular, we construct a bimodule-determined functor Mod Z → Mod R, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continuous surjection Spec Z → Spec R. Algebras satisfying the given axiomatic framework include PI algebras finitely generated over fields, noetherian PI algebras, enveloping algebras of complex finite dimensional solvable Lie algebras, standard generic quantum semisimple Lie groups, quantum affine spaces, quantized Weyl algebras, and standard generic quantizations of the coordinate ring of n × n matrices. In all of these examples (except for the non-finitely-generated noetherian PI algebras), Z is finitely generated over a field, and the constructed map of spectra restricts to a surjection Max Z → Prim R.
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Dissertations / Theses on the topic "Noncommutative algebras"

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Rennie, Adam Charles. "Noncommutative spin geometry." Title page, contents and introduction only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phr4163.pdf.

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Hartman, Gregory Neil. "Graphs and Noncommutative Koszul Algebras." Diss., Virginia Tech, 2002. http://hdl.handle.net/10919/27156.

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A new connection between combinatorics and noncommutative algebra is established by relating a certain class of directed graphs to noncommutative Koszul algebras. The directed graphs in this class are called full graphs and are defined by a set of criteria on the edges. The structural properties of full graphs are studied as they relate to the edge criteria. A method is introduced for generating a Koszul algebra Lambda from a full graph G. The properties of Lambda are examined as they relate to the structure of G, with special attention being given to the construction of a projective resolution of certain semisimple Lambda-modules based on the structural properties of G. The characteristics of the Koszul algebra Lambda that is derived from the product of two full graphs G' and G' are studied as they relate to the properties of the Koszul algebras Lambda' and Lambda' derived from G' and G'.
Ph. D.
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Schoenecker, Kevin J. "An infinite family of anticommutative algebras with a cubic form." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1187185559.

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Russell, Ewan. "Prime ideals in quantum algebras." Thesis, University of Edinburgh, 2009. http://hdl.handle.net/1842/3450.

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The central objects of study in this thesis are quantized coordinate algebras. These algebras originated in the 1980s in the work of Drinfeld and Jumbo and are noncommutative analogues of coordinate rings of algebraic varieties. The organic nature by which these algebras arose is of great interest to algebraists. In particular, investigating ring theoretic properties of these noncommutative algebras in comparison to the properties already known about their classical (commutative) counterparts proves to be a fruitful process. The prime spectrum of an algebra has always been seen as an important key to understanding its fundamental structure. The search for prime spectra is a central focus of this thesis. Our focus is mainly on Quantum Grassmannian subalgebras of quantized coordinate rings of Matrices of size m x n (denoted Oq(Mm;n)). Quantum Grassmannians of size m x n are denoted Gq(m; n) and are the subalgebras generated by the maximal quantum minors of Oq(Mm;n). In Chapter 2 we look at the simplest interesting case, namely the 2 x 4 Quantum Grassmannian (Gq(2; 4)), and we identify the H-primes and automorphism group of this algebra. Chapter 3 begins with a very important result concerning the dehomogenisation isomorphism linking Gq(m; n) and Oq(Mm;n¡m). This result is applied to help to identify H-prime spectra of Quantum Grassmannians. Chapter 4 focuses on identifying the number of H-prime ideals in the 2xn Quan- tum Grassmannian. We show the link between Cauchon fillings of subpartitions and H-prime ideals. In Chapter 5, we look at methods of ordering the generating elements of Quantum Grassmannians and prove the result that Quantum Grassmannians are Quantum Graded Algebras with a Straightening Law is maintained on using one of these alternative orderings. Chapter 6 looks at the Poisson structure on the commutative coordinate ring, G(2; 4) encoded by the noncommutative quantized algebra Gq(2; 4). We describe the symplectic ideals of G(2; 4) based on this structure. Finally in Chapter 7, we present an analysis of the 2 x 2 Reflection Equation Algebra and its primes. This algebra is obtained from the quantized coordinate ring of 2 x 2 matrices, Oq(M2;2).
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Phan, Christopher Lee 1980. "Koszul and generalized Koszul properties for noncommutative graded algebras." Thesis, University of Oregon, 2009. http://hdl.handle.net/1794/10367.

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xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We investigate some homological properties of graded algebras. If A is an R -algebra, then E (A) := Ext A ( R, R ) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré-Birkhoff-Witt deformations. Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality. It is well-known that Poincaré-Birkhoff-Witt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted]. This dissertation contains both previously published and co-authored materials.
Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics
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Meyer, Jonas R. "Noncommutative Hardy algebras, multipliers, and quotients." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/712.

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The principal objects of study in this thesis are the noncommutative Hardy algebras introduced by Muhly and Solel in 2004, also called simply ``Hardy algebras,'' and their quotients by ultraweakly closed ideals. The Hardy algebras form a class of nonselfadjoint dual operator algebras that generalize the classical Hardy algebra, the noncommutative analytic Toeplitz algebras introduced by Popescu in 1991, and other classes of operator algebras studied in the literature. It is known that a quotient of a noncommutative analytic Toeplitz algebra by a weakly closed ideal can be represented completely isometrically as the compression of the algebra to the complement of the range of the ideal, but there is no known general extension of this result to Hardy algebras. An analogous problem on representing quotients of Hardy algebras as compressions of images of induced representations is considered in Chapter 2. Using Muhly and Solel's generalization of Beurling's theorem together with factorizations of weakly continuous linear functionals on infinite multiplicity operator spaces, it is shown that compressing onto the complement of the range of an ultraweakly closed ideal in the space of an infinite multiplicity induced representation yields a completely isometric isomorphism of the quotient. A generalization of Pick's interpolation theorem for elements of Hardy algebras evaluated on their spaces of representations was proved by Muhly and Solel. In Chapter 3, a general theory of reproducing kernel W*-correspondences and their multipliers is developed, generalizing much of the classical theory of reproducing kernel Hilbert space. As an application, it is shown using the generalization of Pick's theorem that the function space representation of a Hardy algebra is isometrically isomorphic (with its quotient norm) to the multiplier algebra of a reproducing kernel W*-correspondence constructed from a generalization of the Szegõ kernel on the unit disk. In Chapter 4, properties of polynomial approximation and analyticity of these functions are studied, with special attention given to the noncommutative analytic Toeplitz algebras. In the final chapter, the canonical curvatures for a class of Hermitian holomorphic vector bundles associated with a C*-correspondence are computed. The Hermitian metrics are closely related to the generalized Szegõ kernels, and when specialized to the disk, the bundle is the Cowen-Douglas bundle associated with the backward shift operator.
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Uhl, Christine. "Quantum Drinfeld Hecke Algebras." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc862764/.

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Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.
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Zhao, Xiangui. "Groebner-Shirshov bases in some noncommutative algebras." London Mathematical Society, 2014. http://hdl.handle.net/1993/24315.

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Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings. This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5). In Chapter 1, we introduce the background and the goals of the thesis. In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp. In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras. In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm. Finally, in Chapter 5, we summarize our results and propose possible future work.
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Oblomkov, Alexei. "Double affine Hecke algebras and noncommutative geometry." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/31165.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.
Includes bibliographical references (p. 93-96).
In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t,tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras.
by Alexei Oblomkov.
Ph.D.
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Gohm, Rolf. "Noncommutative stationary processes /." Berlin [u.a.] : Springer, 2004. http://www.loc.gov/catdir/enhancements/fy0813/2004103932-d.html.

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Books on the topic "Noncommutative algebras"

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Farb, Benson. Noncommutative algebra. New York: Springer-Verlag, 1993.

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Marubayashi, Hidetoshi. Prime Divisors and Noncommutative Valuation Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.

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Khalkhali, Masoud, and Guoliang Yu. Perspectives on noncommutative geometry. Providence, R.I: American Mathematical Society, 2011.

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Silva, Ana Cannas da. Geometric models for noncommutative algebras. Providence, R.I: American Mathematical Society, 1999.

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Rosenberg, Alex. Noncommutative algebraic geometry and representations of quantized algebras. Dordrecht: Kluwer Academic Publishers, 1995.

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Cuculescu, I. Noncommutative probability. Dordrecht: Kluwer Academic Publishers, 1994.

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Rosenberg, Alexander L. Noncommutative Algebraic Geometry and Representations of Quantized Algebras. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8430-2.

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Diep, Do Ngoc. Methods of noncommutative geometry for group C*-algebras. Boca Raton: Chapman & Hall/CRC, 2000.

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Bonfiglioli, Andrea. Topics in noncommutative algebra: The theorem of Campbell, Baker, Hausdorff and Dynkin. Heidelberg: Springer, 2012.

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Doran, Robert S., and Richard V. Kadison, eds. Operator Algebras, Quantization, and Noncommutative Geometry. Providence, Rhode Island: American Mathematical Society, 2004. http://dx.doi.org/10.1090/conm/365.

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Book chapters on the topic "Noncommutative algebras"

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Cuculescu, I., and A. G. Oprea. "Jordan Algebras." In Noncommutative Probability, 293–315. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-015-8374-9_7.

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Arzumanian, Victor, and Suren Grigorian. "Noncommutative Uniform Algebras." In Linear Operators in Function Spaces, 101–9. Basel: Birkhäuser Basel, 1990. http://dx.doi.org/10.1007/978-3-0348-7250-8_5.

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Cuculescu, I., and A. G. Oprea. "Probability on von Neumann Algebras." In Noncommutative Probability, 53–94. Dordrecht: Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-015-8374-9_2.

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Rosenberg, Alexander L. "Noncommutative Affine Schemes." In Noncommutative Algebraic Geometry and Representations of Quantized Algebras, 1–47. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8430-2_1.

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Rosenberg, Alexander L. "Noncommutative Local Algebra." In Noncommutative Algebraic Geometry and Representations of Quantized Algebras, 110–41. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8430-2_3.

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Rosenberg, Alexander L. "Noncommutative Projective Spectrum." In Noncommutative Algebraic Geometry and Representations of Quantized Algebras, 276–305. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8430-2_7.

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Aschieri, Paolo. "Quantum Groups, Quantum Lie Algebras, and Twists." In Noncommutative Spacetimes, 111–32. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-89793-4_7.

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Bratteli, Ola. "Noncommutative vectorfields." In Derivations, Dissipations and Group Actions on C*-algebras, 34–240. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/bfb0098820.

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Gracia-Bondía, José M., Joseph C. Várilly, and Héctor Figueroa. "Kreimer-Connes-Moscovici Algebras." In Elements of Noncommutative Geometry, 597–640. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0005-5_14.

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Várilly, Joseph C. "The Interface of Noncommutative Geometry and Physics." In Clifford Algebras, 227–42. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-1-4612-2044-2_15.

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Conference papers on the topic "Noncommutative algebras"

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VÁRILLY, JOSEPH C. "HOPF ALGEBRAS IN NONCOMMUTATIVE GEOMETRY." In Proceedings of the Summer School. WORLD SCIENTIFIC, 2003. http://dx.doi.org/10.1142/9789812705068_0001.

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Schauenburg, P. "Weak Hopf algebras and quantum groupoids." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-12.

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Khalkhali, M., and B. Rangipour. "Cyclic cohomology of (extended) Hopf algebras." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-5.

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Gomez, X., and S. Majid. "Relating quantum and braided Lie algebras." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-6.

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Szymański, Wojciech. "Quantum lens spaces and principal actions on graph C*-algebras." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-18.

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MORI, IZURU. "NONCOMMUTATIVE PROJECTIVE SCHEMES AND POINT SCHEMES." In Proceedings of the International Conference on Algebras, Modules and Rings. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774552_0014.

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Majewski, Władysław A., and Marcin Marciniak. "On the structure of positive maps between matrix algebras." In Noncommutative Harmonic Analysis with Applications to Probability. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2007. http://dx.doi.org/10.4064/bc78-0-18.

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Wakui, Michihisa. "The coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-20.

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9

LONGO, ROBERTO. "OPERATOR ALGEBRAS AND NONCOMMUTATIVE GEOMETRIC ASPECTS IN CONFORMAL FIELD THEORY." In XVIth International Congress on Mathematical Physics. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814304634_0008.

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Fernández, David, and Luis Álvarez–cónsul. "Noncommutative bi-symplectic $\mathbb{N}Q$-algebras of weight 1." In The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Madrid, Spain). American Institute of Mathematical Sciences, 2015. http://dx.doi.org/10.3934/proc.2015.0019.

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