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Journal articles on the topic 'Base semisimple'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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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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2

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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3

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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4

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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5

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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6

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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7

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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9

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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10

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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11

Ernst, Thomas. "Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces." Special Matrices 9, no. 1 (January 1, 2021): 119–48. http://dx.doi.org/10.1515/spma-2020-0129.

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Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples S U q ( 1 , 1 ) S O q ( 2 ) {{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and S O q ( 3 ) S O q ( 2 ) {{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).

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12

Plymen, R. J. "REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples." Bulletin of the London Mathematical Society 21, no. 2 (March 1989): 202–3. http://dx.doi.org/10.1112/blms/21.2.202.

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13

JAFARIZADEH, M. A., F. NAZERI, and A. KESHISHI. "CALCULATION OF MADELUNG CONSTANT OF VARIOUS IONIC STRUCTURES BASED ON THE SEMISIMPLE LIE ALGEBRAS." Modern Physics Letters B 10, no. 11 (May 10, 1996): 475–85. http://dx.doi.org/10.1142/s0217984996000523.

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Periodic structures are constructed by using semisimple Lie algebras. Then these structures are generalized to ionic structures. Finally Madelung constant of these structures are calculated by solving Poisson equation in the fundamental domain of corresponding Lie algebras with Dirichlet boundary conditions. The obtained results are consistent with data of the ionic crystals with the same coordination numbers as our structures.

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14

SFETSOS, KONSTADINOS. "EXACT STRING BACKGROUNDS FROM WZW MODELS BASED ON NON-SEMISIMPLE GROUPS." International Journal of Modern Physics A 09, no. 27 (October 30, 1994): 4759–66. http://dx.doi.org/10.1142/s0217751x94001916.

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We formulate WZW models based on a centrally extended version of the Euclidean group in d dimensions. We obtain string backgrounds corresponding to conformal σ models in D=d2 space-time dimensions with exact central charge c=d2 and d(d−1)/2 null Killing vectors. By identifying the corresponding conformal field theory we show that the one-loop results coincide with the exact ones up to a shifting of a parameter.

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15

Vogan Jr., David A. "Book Review: Representation theory of semisimple groups. An overview based on examples." Bulletin of the American Mathematical Society 17, no. 2 (October 1, 1987): 392–97. http://dx.doi.org/10.1090/s0273-0979-1987-15612-6.

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16

Sogami, I. S. "Sum Rules for Elements of Flavor-Mixing Matrices Based on a Non-Semisimple Symmetry." Progress of Theoretical Physics 115, no. 2 (February 1, 2006): 461–65. http://dx.doi.org/10.1143/ptp.115.461.

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17

Dong, Qixiang. "Projection-based commuting solutions of the Yang–Baxter matrix equation for non-semisimple eigenvalues." Applied Mathematics Letters 64 (February 2017): 231–34. http://dx.doi.org/10.1016/j.aml.2016.09.013.

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18

Nikmehr, M. J., and S. Khojasteh. "A generalized ideal-based zero-divisor graph." Journal of Algebra and Its Applications 14, no. 06 (April 21, 2015): 1550079. http://dx.doi.org/10.1142/s0219498815500796.

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Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

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19

Edwards, S. A., and M. D. Gould. "A projection based approach to the Clebsch-Gordan multiplicity problem for compact semisimple Lie groups: I. General formalism." Journal of Physics A: Mathematical and General 19, no. 9 (June 21, 1986): 1523–29. http://dx.doi.org/10.1088/0305-4470/19/9/022.

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20

Gould, M. D., and S. A. Edwards. "A projection based approach to the Clebsch-Gordan multiplicity problem for compact semisimple Lie groups. III. The classical limit." Journal of Physics A: Mathematical and General 19, no. 9 (June 21, 1986): 1537–44. http://dx.doi.org/10.1088/0305-4470/19/9/024.

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21

Edwards, S. A., and M. D. Gould. "A projection based approach to the Clebsch-Gordan multiplicity problem for compact semisimple Lie groups. II. Application to U(n)." Journal of Physics A: Mathematical and General 19, no. 9 (June 21, 1986): 1531–36. http://dx.doi.org/10.1088/0305-4470/19/9/023.

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22

JONES, PETER R. "THE SEMIGROUPS B2 AND B0 ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS." International Journal of Algebra and Computation 23, no. 06 (September 2013): 1289–335. http://dx.doi.org/10.1142/s0218196713500264.

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The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids". These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.

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