The Experts below are selected from a list of 78 Experts worldwide ranked by ideXlab platform
N. A. Koreshkov - One of the best experts on this subject based on the ideXlab platform.
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Lie Algebras and Algebras of associative type
Mathematical Notes, 2010Co-Authors: N. A. KoreshkovAbstract:In the paper, some properties of Algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional Semisimple modular Lie Algebras. It is proved that the homogeneous radical of any finite-dimensional Algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional Semisimple Algebra of associative type A = ⊕_ α ε G A _ α graded by some group G , over a field of characteristic zero, has a nonzero component A _1 (where 1 stands for the identity element of G ), and A _1 is a Semisimple associative Algebra. Let B = ⊕_ α ε G B _ α be a finite-dimensional Semisimple Lie Algebra over a prime field F _ p , and let B be graded by a commutative group G . If B = F _ p ⊗_ ℤ A _ L , where A _ L is the commutator Algebra of a ℤ-Algebra A = ⊕_ α ε G A _ α ; if ℚ ◯ _ ℤ A is an Algebra of associative type, then the 1-component of the Algebra K ◯ _ ℤ B , where K stands for the Algebraic closure of the field F _ p , is the sum of some Algebras of the form gl( n _ i , K ).
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Modules and ideals of Algebras of associative type
Russian Mathematics, 2008Co-Authors: N. A. KoreshkovAbstract:In this paper, we study some properties of Algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional Algebra of associative type over a field of zero characteristic is homogeneously Semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of Algebras of associative type, it is proved that any module over a Semisimple Algebra is completely reducible. We also prove that any left homogeneous ideal of a Semisimple Algebra of associative type is generated by a homogeneous idempotent.
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On the nilpotency and decomposition of Lie-type Algebras
Mathematical Notes, 2007Co-Authors: N. A. KoreshkovAbstract:In the paper, an analog of the Engel theorem for graded Algebras admitting a Lie-type module is proved. Moreover, it is shown that every Semisimple Algebra of associative type with ordered grading and one-dimensional grading subspaces is the direct sum of two-sided ideals that are simple Algebras.
Sarah Wolff - One of the best experts on this subject based on the ideXlab platform.
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The Efficient Computation of Fourier Transforms on Semisimple Algebras
Journal of Fourier Analysis and Applications, 2018Co-Authors: David Maslen, Daniel N. Rockmore, Sarah WolffAbstract:We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a Semisimple Algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path Algebra and construction of Gel’fand–Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley–Lieb, and Birman–Murakami–Wenzl Algebras.
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The efficient computation of Fourier transforms on Semisimple Algebras
arXiv: Representation Theory, 2016Co-Authors: David Maslen, Daniel N. Rockmore, Sarah WolffAbstract:We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a Semisimple Algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path Algebra and construction of Gel'fand-Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley-Lieb Algebras, and Birman-Murakami-Wenzl Algebras.
David Maslen - One of the best experts on this subject based on the ideXlab platform.
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The Efficient Computation of Fourier Transforms on Semisimple Algebras
Journal of Fourier Analysis and Applications, 2018Co-Authors: David Maslen, Daniel N. Rockmore, Sarah WolffAbstract:We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a Semisimple Algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path Algebra and construction of Gel’fand–Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley–Lieb, and Birman–Murakami–Wenzl Algebras.
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The efficient computation of Fourier transforms on Semisimple Algebras
arXiv: Representation Theory, 2016Co-Authors: David Maslen, Daniel N. Rockmore, Sarah WolffAbstract:We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a Semisimple Algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path Algebra and construction of Gel'fand-Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley-Lieb Algebras, and Birman-Murakami-Wenzl Algebras.
Laura Năstăsescu - One of the best experts on this subject based on the ideXlab platform.
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Graded Semisimple Algebras are symmetric
Journal of Algebra, 2017Co-Authors: Sorin Dăscălescu, Constantin Năstăsescu, Laura NăstăsescuAbstract:Abstract We study graded symmetric Algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded Semisimple Algebra is graded symmetric. The center of a symmetric Algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division Algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.
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Are graded Semisimple Algebras symmetric
arXiv: Rings and Algebras, 2015Co-Authors: Sorin Dăscălescu, Constantin Năstăsescu, Laura NăstăsescuAbstract:We study graded symmetric Algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded division Algebra whose dimension is not divisible by the characteristic of the base field is graded symmetric. Using the structure of graded simple (Semisimple) Algebras, we extend the results to these classes. In particular, in characteristic zero any graded Semisimple Algebra is graded symmetric. We show that the center of a finite dimensional graded division Algebra is often symmetric.
R. Coquereaux - One of the best experts on this subject based on the ideXlab platform.
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On the Finite-Dimensional Quantum Group M_3⊕(M_2|1(Λ^2))0
Letters in Mathematical Physics, 1997Co-Authors: R. CoquereauxAbstract:We describe a few properties of the nonSemisimple associative Algebra ℋ=M_3 ⊕ (M_2|1 (Λ^2))_0, where Λ^2 is the Grassmann Algebra with two generators. We show that ℋ is not only a finite-dimensional Algebra but also a (noncommutative) Hopf Algebra, hence a finite-dimensional quantum group. By selecting a system of explicit generators, we show how it is related with the quantum enveloping of SL_q(2) when the parameter q is a cubic root of unity. We describe its indecomposable projective representations as well as the irreducible ones. We also comment about the relation between this object and the theory of modular representation of the group SL(2, F_3), i.e. the binary tetrahedral group. Finally, we briefly discuss its relation with the Lorentz group and, as already suggested by A.~Connes, make a few comments about the possible use of this Algebra in a modification of the Standard Model of particle physics (the unitary group of the Semisimple Algebra associated with ℋ is U(3) × U(2) × U(1)).
