Relevant bibliographies by topics / Base semisimple

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

Academic literature on the topic 'Base semisimple'

Author: Grafiati
Published: 4 June 2021
Last updated: 12 February 2022

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Journal articles on the topic "Base semisimple"

1

McDougall, Robert. "The base semisimple class." Communications in Algebra 28, no. 9 (January 2000): 4269–83. http://dx.doi.org/10.1080/00927870008827089.

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MCDOUGALL, R. G., and L. K. THORNTON. "ON BASE RADICAL OPERATORS FOR CLASSES OF FINITE ASSOCIATIVE RINGS." Bulletin of the Australian Mathematical Society 98, no. 2 (July 18, 2018): 239–50. http://dx.doi.org/10.1017/s0004972718000461.

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In this paper, class operators are used to give a complete listing of distinct base radical and semisimple classes for universal classes of finite associative rings. General relations between operators reveal that the maximum order of the semigroup formed is 46. In this setting, the homomorphically closed semisimple classes are precisely the hereditary radical classes and hence radical–semisimple classes, and the largest homomorphically closed subclass of a semisimple class is a radical–semisimple class.
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Okniński, Jan, and Mohan S. Putcha. "Embedding finite semigroup amalgams." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 51, no. 3 (December 1991): 489–96. http://dx.doi.org/10.1017/s1446788700034650.

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AbstractLet S, T1,… Tk be finite semigroups and Ψ: S → Ti, be embeddings. When C[S] is semisimple, we find necessary and sufficient conditions for the semigroup amalgam (T1,…, Tk; S) to be embeddable in a finite semigroup. As a consequence we show that if S is a finite semigroup with C[S] semisimple, then S is an amalgamation base for the class of finite semigroups if and only if the principal ideals of S are linearly ordered. Our proof uses both the theory of representations by transformations and the theory of matrix representations as developed by Clifford, Munn and Ponizovskii
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McConnell, N. R., R. G. McDougall, and T. Stokes. "On base radical and semisimple classes defined by class operators." Acta Mathematica Hungarica 138, no. 4 (August 1, 2012): 307–28. http://dx.doi.org/10.1007/s10474-012-0249-9.

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BICHON, JULIEN. "GALOIS AND BIGALOIS OBJECTS OVER MONOMIAL NON-SEMISIMPLE HOPF ALGEBRAS." Journal of Algebra and Its Applications 05, no. 05 (October 2006): 653–80. http://dx.doi.org/10.1142/s0219498806001934.

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We describe the Hopf–Galois extensions of the base field and the biGalois groups of non-semisimple monomial Hopf algebras. The main feature of our description is the use of modified versions of the second cohomology group of the grouplike elements. Our computations generalize the previous ones of Masuoka and Schauenburg for the Taft algebras.
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Mcconnell, N. R., R. G. Mcdougall, and T. Stokes. "Erratum to: On base radical and semisimple classes defined by class operators." Acta Mathematica Hungarica 144, no. 1 (September 5, 2014): 266–68. http://dx.doi.org/10.1007/s10474-014-0444-y.

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Thornton, L. K. "On base radical and semisimple operators for a class of finite algebras." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 59, no. 2 (December 6, 2017): 361–74. http://dx.doi.org/10.1007/s13366-017-0371-5.

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8

Chaput, Pierre-Emmanuel, and Matthieu Romagny. "On the adjoint quotient of Chevalley groups over arbitrary base schemes." Journal of the Institute of Mathematics of Jussieu 9, no. 4 (April 16, 2010): 673–704. http://dx.doi.org/10.1017/s1474748010000125.

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AbstractFor a split semisimple Chevalley group scheme G with Lie algebra $\mathfrak{g}$ over an arbitrary base scheme S, we consider the quotient of $\mathfrak{g} by the adjoint action of G. We study in detail the structure of $\mathfrak{g} over S. Given a maximal torus T with Lie algebra $\mathfrak{t}$ and associated Weyl group W, we show that the Chevalley morphism π : $\mathfrak{t}$/W → $\mathfrak{g}/G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient $\mathfrak{g} → \mathfrak{g}//G commutes, or does not commute, with base change on S.
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Losev, Ivan, and Victor Ostrik. "Classification of finite-dimensional irreducible modules over -algebras." Compositio Mathematica 150, no. 6 (April 7, 2014): 1024–76. http://dx.doi.org/10.1112/s0010437x13007604.

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AbstractFinite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.
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Zhang, Zhirang, and Xuemei Li. "The Upper Radical Property and Lower Radical Property of Groups." Algebra Colloquium 18, no. 04 (December 2011): 693–700. http://dx.doi.org/10.1142/s100538671100054x.

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We take in this paper an arbitrary class [Formula: see text] of groups as a base, and define a radical property 𝒫 for which every group in [Formula: see text] is 𝒫-semisimple. This is called the upper radical property determined by the class [Formula: see text]. At the same time, we define a radical property 𝒫 for which every group in [Formula: see text] is a 𝒫-radical group. This is called the first lower radical property determined by the class [Formula: see text]. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one.
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Dissertations / Theses on the topic "Base semisimple"

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Chin, Melanie Soo, and m. chin@cqu edu au. "Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions." Central Queensland University. Computer Science, 2004. http://library-resources.cqu.edu.au./thesis/adt-QCQU/public/adt-QCQU20050411.102928.

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This research aims to refresh and reinterpret the radical theory of associative rings using the base radical and base semisimple class constructions. It also endeavours to generalise some results about ideals of rings in terms of accessible subrings. A characterisation of accessible subrings is included. By applying the base radical and base semisimple class constructions to many of the known results in established radical theory a number of gaps are uncovered and closed, with the goal of making the theory more accessible to advanced undergraduate and graduate students and mathematicians in related fields, and to open up new areas of investigation. After a literature review and brief reminder of algebra rudiments, the useful properties of accessible subrings and the U and S operators independent from radical class connections are described. The section on accessible subrings illustrates that replacing ideals with accessible subrings is indeed possible for a number of results and demonstrates its usefulness. The traditional radical and semisimple class definitions are included and it is shown that the base radical and base semisimple class constructions are equivalent. Diagrams illustrating the constructions support the definitions. From then on, all radical and semisimple classes mentioned are understood to have the base radical and base semisimple class form. Subject to the constraints of this work, many known results of traditional radical theory are reinterpreted with new proofs, illustrating the potential to simplify the understanding of radical theory using the base radical and base semisimple class constructions. Along with reinterpreting known results, new results emerge giving further insight to radical theory and its intricacies. Accessible subrings and the U and S operators are integrated into the development. The duality between the base radical and base semisimple class constructions is demonstrated in earnest. With a measure of the theory presented, the new constructions are applied to examples and concrete radicals. Context is supported by establishing the relationship between some well-known rings and the radical and related classes of interest. The title of the thesis, Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions, reflects the understanding that reinterpreting the entirety of radical theory is beyond the scope of this work. The conclusion includes an outlook listing further research that time did not allow.
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Athapattu, Mudiyanselage Chathurika Umayangani Manike Athapattu. "Chevalley Groups." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1986.

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In this thesis, we construct Chevalley groups over arbitrary fields. The construction is based on the properties of semi-simple complex Lie algebras, the existence of Chevalley bases and notion of universal enveloping algebras. Using integral lattices in universal enveloping algebras and integral properties of Chevalley bases, we present a method which produces, for any complex simple Lie group, an analogous group over an arbitrary field.
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Books on the topic "Base semisimple"

1

Knapp, Anthony W. Representation theory of semisimple groups: An overview based on examples. Princeton, N.J: Princeton University Press, 2001.

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Representation theory of semisimple groups, an overview based on examples. Princeton, N.J: Princeton University Press, 1986.

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Knapp, Anthony W. Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton University Press, 2016.

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Gaitsgory, Dennis, and Jacob Lurie. Weil's Conjecture for Function Fields. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691182148.001.0001.

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A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil's conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-adic sheaves. Using this theory, the authors articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck–Lefschetz trace formula, the book shows that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.
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Book chapters on the topic "Base semisimple"

1

Mitzman, David. "3. Chevalley bases for the remaining semisimple and affine Lie algebras." In Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras, 38–110. Providence, Rhode Island: American Mathematical Society, 1985. http://dx.doi.org/10.1090/conm/040/03.

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Mitzman, David. "2. Chevalley bases for semisimple and type 1 affine Lie algebras of types A, D, E." In Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras, 9–37. Providence, Rhode Island: American Mathematical Society, 1985. http://dx.doi.org/10.1090/conm/040/02.

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Bismut, Jean-Michel. "The displacement function and the return map." In Hypoelliptic Laplacian and Orbital Integrals (AM-177). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691151298.003.0004.

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This chapter studies the displacement function dᵧ on X that is associated with a semisimple element γ‎ ∈ G. If φ‎″, t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X(γ‎) ⊂ X of dᵧ can be easily related to the fixed point set Fᵧ ⊂ X of the symplectic transformation γ‎⁻¹φ‎₁ of X. The chapter studies the nondegeneracy of γ‎⁻¹φ‎₁ − 1 along Fᵧ. More fundamentally, this chapter gives important quantitative estimates on how much φ‎ ½ differs from φ‎ ˗½γ‎ away from Fᵧ. These quantitative estimates are based on Toponogov's theorem.
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Journal articles
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