54 results on '"Mikhail Kochetov"'
Search Results
2. On nonassociative graded-simple algebras over the field of real numbers
- Author
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Mikhail Kochetov and Yuri Bahturin
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Pure mathematics ,Loop algebra ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,Field (mathematics) ,Mathematics - Rings and Algebras ,0102 computer and information sciences ,01 natural sciences ,Real closed field ,16W50 (Primary) 17D05, 17C20, 17C60 (Secondary) ,Rings and Algebras (math.RA) ,010201 computation theory & mathematics ,Simple (abstract algebra) ,FOS: Mathematics ,Isomorphism ,0101 mathematics ,Abelian group ,Equivalence (measure theory) ,Real number ,Mathematics - Abstract
We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence., Comment: 24 pages
- Published
- 2019
3. Gradings on associative algebras with involution and real forms of classical simple Lie algebras
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Adrián Rodrigo-Escudero, Alberto Elduque, and Mikhail Kochetov
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,Field (mathematics) ,Mathematics - Rings and Algebras ,02 engineering and technology ,16. Peace & justice ,01 natural sciences ,Real closed field ,17B70 (Primary) 16W50, 16W10 (Secondary) ,Simple (abstract algebra) ,Rings and Algebras (math.RA) ,Lie algebra ,FOS: Mathematics ,Group homomorphism ,0101 mathematics ,Abelian group ,Associative property ,Mathematics ,Real number - Abstract
We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras., Comment: 61 pages. To appear in J. Algebra
- Published
- 2021
- Full Text
- View/download PDF
4. Group Gradings on Finitary Simple Lie Algebras.
- Author
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Yuri A. Bahturin, Matej Bresar, and Mikhail Kochetov
- Published
- 2012
- Full Text
- View/download PDF
5. Graded-division algebras and Galois extensions
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Mikhail Kochetov and Alberto Elduque
- Subjects
Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,Galois theory ,Galois group ,Mathematics - Rings and Algebras ,01 natural sciences ,Primary 16W50, Secondary 16K20, 16K50 ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Isomorphism class ,0101 mathematics ,Abelian group ,Algebra over a field ,Brauer group ,Quotient ,Mathematics - Abstract
Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras. On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field F is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of F and the isomorphism class of a G-Galois extension of F. This connection is used to classify the simple G-Galois extensions of F in terms of a Galois field extension L/F with Galois group isomorphic to a quotient G/K and the class of a 2-cocycle of K with values in the multiplicative group of L modulo a 2-coboundary with values in the multiplicative group of F, subject to certain conditions. Non-simple G-Galois extensions are induced from simple T-Galois extensions for a subgroup T of G. We also classify finite-dimensional G-graded-division algebras and, as an application, finite G-graded-division rings., Comment: 33 pages
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- 2020
- Full Text
- View/download PDF
6. Gradings on Modules Over Lie Algebras of E Types
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Mikhail Kochetov, Cristina Draper, and Alberto Elduque
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,16W50 (Primary), 17B70, 17B25, 17B60 (Secondary) ,Mathematics - Rings and Algebras ,01 natural sciences ,Rings and Algebras (math.RA) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Abelian group ,Algebraically closed field ,Mathematics - Representation Theory ,Mathematics - Abstract
For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of $G$-graded simple $\mathcal{L}$-modules, as well as necessary and sufficient conditions for an $\mathcal{L}$-module to admit a $G$-grading compatible with the given $G$-grading on $\mathcal{L}$., 21 pages
- Published
- 2017
7. Group gradings on the Lie and Jordan superalgebrasQ(n)
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Caio De Naday Hornhardt, Helen Samara Dos Santos, Yuri Bahturin, and Mikhail Kochetov
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Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,Simple (abstract algebra) ,Mathematics::Rings and Algebras ,010102 general mathematics ,Graded ring ,010103 numerical & computational mathematics ,0101 mathematics ,Abelian group ,Algebraically closed field ,01 natural sciences ,Mathematics - Abstract
We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras Q(n), n ≥ 2, over an algebraically closed field of characteristic different from 2 (and not div...
- Published
- 2016
8. Graded-division algebras over arbitrary fields
- Author
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Alberto Elduque, Yuri Bahturin, and Mikhail Kochetov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Group (mathematics) ,Applied Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,0211 other engineering and technologies ,Graded ring ,021107 urban & regional planning ,Field (mathematics) ,02 engineering and technology ,01 natural sciences ,Real closed field ,Field extension ,Division algebra ,Galois extension ,0101 mathematics ,Abelian group ,Mathematics - Abstract
A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field [Formula: see text] can be reduced to the following three classifications, for each finite Galois extension [Formula: see text] of [Formula: see text]: (1) finite-dimensional central division algebras over [Formula: see text], up to isomorphism; (2) twisted group algebras of finite groups over [Formula: see text], up to graded-isomorphism; (3) [Formula: see text]-forms of certain graded matrix algebras with coefficients in [Formula: see text] where [Formula: see text] is as in (1) and [Formula: see text] is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.
- Published
- 2020
9. Group gradings on the Lie and Jordan algebras of block-triangular matrices
- Author
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Felipe Yukihide Yasumura and Mikhail Kochetov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Jordan algebra ,010102 general mathematics ,17B70 (Primary), 16W50, 17C99 (secondary) ,Mathematics - Rings and Algebras ,01 natural sciences ,Rings and Algebras (math.RA) ,0103 physical sciences ,Associative algebra ,Lie algebra ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraically closed field ,Mathematics - Abstract
We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading always generates an abelian subgroup of $G$. Assuming that $G$ is abelian, our technique also works to obtain the classification of $G$-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic $0$ and $G$ finite abelian) and classified up to isomorphism by A. Borges et al. in 2018. Finally, still assuming that $G$ is abelian, we classify $G$-gradings on the upper block-triangular matrices as a Jordan algebra, over an algebraically closed field of characteristic $0$. It turns out that, under these assumptions, the Jordan case is equivalent to the Lie case.
- Published
- 2018
- Full Text
- View/download PDF
10. Graded modules over classical simple Lie algebras with a grading
- Author
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Alberto Elduque and Mikhail Kochetov
- Subjects
Pure mathematics ,Primary 17B70, Secondary 17B10, 16W50 ,General Mathematics ,Mathematics::Rings and Algebras ,Outer automorphism group ,Mathematics - Rings and Algebras ,Rings and Algebras (math.RA) ,Lie algebra ,FOS: Mathematics ,Isomorphism class ,Representation Theory (math.RT) ,Abelian group ,Algebraically closed field ,Grading (education) ,Semisimple Lie algebra ,Mathematics - Representation Theory ,Mathematics - Abstract
Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module., 38 pages
- Published
- 2015
11. Effective Elasticity Tensors in Context of Random Errors
- Author
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Mikhail Kochetov, Michael A. Slawinski, and Tomasz Danek
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Tensor contraction ,Cartesian tensor ,Mechanics of Materials ,Cauchy stress tensor ,Mechanical Engineering ,Compatibility (mechanics) ,Mathematical analysis ,Symmetric tensor ,General Materials Science ,Tensor ,Tensor density ,Mathematics ,Tensor field - Abstract
We introduce the effective elasticity tensor of a chosen material-symmetry class to represent a measured generally anisotropic elasticity tensor by minimizing the weighted Frobenius distance from the given tensor to its symmetric counterpart, where the weights are determined by the experimental errors. The resulting effective tensor is the highest-likelihood estimate within the specified symmetry class. Given two material-symmetry classes, with one included in the other, the weighted Frobenius distance from the given tensor to the two effective tensors can be used to decide between the two models—one with higher and one with lower symmetry—by means of the likelihood ratio test.
- Published
- 2015
12. Gradings by Groups on Cartan Type Lie Algebras
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Yuri Bahturin, Mikhail Kochetov, and Jason McGraw
- Published
- 2015
13. Gradings on classical central simple real Lie algebras
- Author
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Adrián Rodrigo-Escudero, Yuri Bahturin, and Mikhail Kochetov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Simple Lie group ,010102 general mathematics ,Non-associative algebra ,17B70 (Primary), 16W50, 16W10 (Secondary) ,Real form ,010103 numerical & computational mathematics ,Mathematics - Rings and Algebras ,Killing form ,01 natural sciences ,Affine Lie algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Fundamental representation ,0101 mathematics ,Mathematics - Abstract
For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field)., 35 pages
- Published
- 2017
14. Group gradings on the superalgebras M(m,n), A(m,n) and P(n)
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Mikhail Kochetov, Helen Samara Dos Santos, and Caio De Naday Hornhardt
- Subjects
Algebra and Number Theory ,Mathematics::Commutative Algebra ,010102 general mathematics ,Mathematics::Rings and Algebras ,Lie superalgebra ,Mathematics - Rings and Algebras ,16. Peace & justice ,01 natural sciences ,Combinatorics ,Rings and Algebras (math.RA) ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraically closed field ,Mathematics::Representation Theory ,17B70 (Primary) 16W50, 16W55, 17A70 (Secondary) ,Mathematics - Abstract
We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n \geq 2$, and on the simple associative superalgebras $M(m,n)$, $m, n \geq 1$, over an algebraically closed field: fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism. As a corollary, we also classify up to isomorphism the $G$-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from $G$-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be $0$., 29 pages
- Published
- 2017
15. Uncertainty analysis of effective elasticity tensors using quaternion-based global optimization and Monte-Carlo method
- Author
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Mikhail Kochetov, Tomasz Danek, and Michael A. Slawinski
- Subjects
Mathematical optimization ,Mechanics of Materials ,Applied Mathematics ,Mechanical Engineering ,Monte Carlo method ,Applied mathematics ,Elasticity (economics) ,Condensed Matter Physics ,Quaternion ,Global optimization ,Uncertainty analysis ,Mathematics - Published
- 2013
16. Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities
- Author
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Mikhail Kochetov, Alexey Gordienko, Mathematics, and Algebra
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Discrete mathematics ,Wedderburn - Mal'cev Theorem ,Pure mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,derivation ,grading ,Universal enveloping algebra ,codimension ,16. Peace & justice ,Representation theory ,Affine Lie algebra ,Hopf algebra ,Graded Lie algebra ,Levi Theorem ,Lie algebra ,Division algebra ,Algebra representation ,polynomial identity ,affine algebraic group ,H-module algebra ,associative algebra ,Lie Algebra ,Semisimple Lie algebra ,Mathematics - Abstract
Suppose a finite dimensional semisimple Lie algebra \(\mathfrak g\) acts by derivations on a finite dimensional associative or Lie algebra A over a field of characteristic 0. We prove the \(\mathfrak g\)-invariant analogs of Wedderburn—Mal’cev and Levi theorems, and the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of A. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur’s conjecture for finite dimensional associative algebras with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms or graded by an arbitrary Abelian group. In addition, we provide criteria for G-, H- and graded simplicity in terms of codimensions.
- Published
- 2013
17. Graded simple modules and loop modules
- Author
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Alberto Elduque and Mikhail Kochetov
- Subjects
010102 general mathematics ,0103 physical sciences ,Mathematics::Rings and Algebras ,FOS: Mathematics ,010307 mathematical physics ,16W50 (Primary), 17B70 (Secondary) ,0101 mathematics ,Representation Theory (math.RT) ,01 natural sciences ,Mathematics - Representation Theory - Abstract
Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be isomorphic to each other. Under some restrictions, these loop modules are completely reducible (as ungraded modules), and some of their invariants --- inertia group, graded Brauer invariant and Schur index --- which were previously defined for simple modules over graded finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero, are now considered in a more general and natural setting., 30 pages; some explanatory remarks are added in this version
- Published
- 2016
18. Gradings on the exceptional Lie algebras $F_4$ and $G_2$ revisited
- Author
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Mikhail Kochetov and Alberto Elduque
- Subjects
Weyl group ,Pure mathematics ,Jordan algebra ,Primary 17B70, secondary 17D05, 17C40 ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Albert algebra ,Graded ring ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Rings and Algebras (math.RA) ,Simple (abstract algebra) ,Lie algebra ,FOS: Mathematics ,symbols ,0101 mathematics ,Algebraically closed field ,Abelian group ,Mathematics - Abstract
All gradings by abelian groups are classified on the following algebras over an algebraically closed field of characteristic not 2: the simple Lie algebra of type $G_2$ (characteristic not 3), the exceptional simple Jordan algebra, and the simple Lie algebra of type $F_4$., 36 pages. References expanded and updated
- Published
- 2012
19. Group gradings on restricted Cartan-type Lie algebras
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Yuri Bahturin and Mikhail Kochetov
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Pure mathematics ,Simple (abstract algebra) ,Group (mathematics) ,General Mathematics ,Lie algebra ,Isomorphism ,Abelian group ,Type (model theory) ,Algebraically closed field ,Mathematics ,Ground field - Abstract
For a given abelian group G, we classify the isomorphism classes of Ggradings on the simple restricted Lie algebras of types W.mI1/ and S.mI1/ for m 2, in terms of numerical and group-theoretical invariants. Our main tool is automorphism group schemes, which we determine for the simple restricted Lie algebras of types S.mI1/ and H.mI1/. The ground field is assumed to be algebraically closed of characteristic p > 3.
- Published
- 2011
20. Classification of group gradings on simple Lie algebras of types A, B, C and D
- Author
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Yuri Bahturin and Mikhail Kochetov
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Simple Lie group ,010102 general mathematics ,Non-associative algebra ,Elementary abelian group ,010103 numerical & computational mathematics ,01 natural sciences ,Affine Lie algebra ,Representation theory ,Lie conformal algebra ,Combinatorics ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,0101 mathematics ,Mathematics - Abstract
For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple Lie algebras of types A n ( n ⩾ 1 ), B n ( n ⩾ 2 ), C n ( n ⩾ 3 ) and D n ( n > 4 ), in terms of numerical and group-theoretical invariants. The ground field is assumed to be algebraically closed of characteristic different from 2.
- Published
- 2010
21. On Choosing Effective Symmetry Classes for Elasticity Tensors
- Author
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Michael A. Slawinski, Mikhail Kochetov, and Çağri Diner
- Subjects
Tensor contraction ,Weyl tensor ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Condensed Matter Physics ,Tensor field ,symbols.namesake ,Theoretical physics ,Exact solutions in general relativity ,Cartesian tensor ,Mechanics of Materials ,symbols ,Symmetric tensor ,Tensor ,Tensor density ,Mathematics - Abstract
We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective tensor is based on examining the features of the plot of the monoclinic-distance function of a given tensor, choosing an appropriate symmetry class, and then finding the closest tensor in that class. The concept of the effective tensor is not tantamount to the closest tensor since one always obtains a closer approximation using a monoclinic tensor than a tensor of any other nontrivial symmetry. Hence, we use qualitative features of the plot of the monoclinic-distance function to choose an effective symmetry class within which the closest tensor can be computed.
- Published
- 2010
22. Identifying Symmetry Classes of Elasticity Tensors Using Monoclinic Distance Function
- Author
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Michael A. Slawinski, Çağri Diner, and Mikhail Kochetov
- Subjects
Unit sphere ,Mechanical Engineering ,Coordinate system ,Mathematical analysis ,Geometry ,Euler angles ,symbols.namesake ,Mechanics of Materials ,Transverse isotropy ,symbols ,General Materials Science ,Tensor ,Elasticity (economics) ,Anisotropy ,Mathematics ,Monoclinic crystal system - Abstract
We present a method to identify the symmetry class of an elasticity tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the concept of distance in the space of tensors, and relies on the monoclinic or transversely isotropic distance function. Since the orientation of a monoclinic or transversely isotropic tensor depends on two Euler angles only, we can plot the corresponding distance functions on the unit sphere in ℝ3 and observe the symmetry pattern of the plot. In particular, the monoclinic distance function vanishes in the directions of the normals of the mirror planes, so the number and location of the zeros allows us to identify the symmetry class and the orientation of the natural coordinate system. Observing the approximate locations of the zeros on the plot, we can constrain a numerical algorithm for finding the exact orientation of the natural coordinate system.
- Published
- 2010
23. Estimating effective elasticity tensors from Christoffel equations
- Author
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Michael A. Slawinski and Mikhail Kochetov
- Subjects
Weyl tensor ,Tensor contraction ,Mathematical analysis ,Tensor field ,symbols.namesake ,Geophysics ,Exact solutions in general relativity ,Cartesian tensor ,Geochemistry and Petrology ,symbols ,Symmetric tensor ,Tensor ,Tensor density ,Mathematics - Abstract
We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities. These quantities — the wavefront-slowness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization problem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of particular symmetry without assuming its orientation is challenging. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we distinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.
- Published
- 2009
24. Group gradings on the Lie algebra psln in positive characteristic
- Author
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Yuri Bahturin and Mikhail Kochetov
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Simple Lie group ,010102 general mathematics ,Adjoint representation ,Universal enveloping algebra ,Lie superalgebra ,01 natural sciences ,Affine Lie algebra ,Graded Lie algebra ,Lie conformal algebra ,010101 applied mathematics ,Filtered algebra ,0101 mathematics ,Mathematics - Abstract
In this paper we describe all gradings by an abelian group G on the simple Lie algebra p s l n ( F ) where F is an algebraically closed field of characteristic p different from 2 and dividing n .
- Published
- 2009
25. On obtaining effective orthotropic elasticity tensors
- Author
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Mikhail Kochetov and Michael A. Slawinski
- Subjects
Cauchy stress tensor ,Applied Mathematics ,Mechanical Engineering ,Numerical analysis ,Mathematical analysis ,Matrix norm ,Geometry ,Condensed Matter Physics ,Orthotropic material ,symbols.namesake ,Mechanics of Materials ,Transverse isotropy ,symbols ,Tensor ,Frobenius theorem (differential topology) ,Rotation group SO ,Mathematics - Abstract
We consider the problem of obtaining the effective orthotropic tensor that corresponds to a given generally anisotropic one; herein, by 'effective', we mean the closest in the sense of the Frobenius norm, without a priori assuming the orientation of the orthotropic tensor. It is difficult to find the absolute minimum of the distance function since the minimization process is nonlinear, exhibiting several local minima. To find the effective orthotropic tensor, the minimization process must be performed on a three-dimensional manifold SO(3). In the case of monoclinic and transversely isotropic tensors, it can be performed on a two-dimensional sphere, which lends itself to an insightful plot that allows us to guide a numerical method. We use the orientation of the symmetry-plane normal of the effective monoclinic tensor to guide the method and obtain the effective orthotropic tensor-a two-step process.
- Published
- 2009
26. Gradings on Finite-Dimensional Simple Lie Algebras
- Author
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Mikhail Kochetov
- Subjects
Algebra ,Adjoint representation of a Lie algebra ,Pure mathematics ,Applied Mathematics ,Simple Lie group ,Non-associative algebra ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Representation theory ,Mathematics ,Lie conformal algebra - Abstract
In this survey paper we present recent classification results for gradings by arbitrary groups on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from 2. We also describe the main tools that were used to obtain these results (in particular, the classification of group gradings on matrix algebras).
- Published
- 2008
27. Generalized Lie Algebras and Cocycle Twists
- Author
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Mikhail Kochetov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Non-associative algebra ,Killing form ,Affine Lie algebra ,Graded Lie algebra ,Lie conformal algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Freudenthal magic square ,Mathematics - Abstract
We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.
- Published
- 2008
28. On Obtaining Effective Transversely Isotropic Elasticity Tensors
- Author
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Mikhail Kochetov and Michael A. Slawinski
- Subjects
Optimization problem ,Mechanical Engineering ,Mathematical analysis ,Matrix norm ,Geometry ,Maxima and minima ,Mechanics of Materials ,Transverse isotropy ,General Materials Science ,Tensor ,Elasticity (economics) ,Quaternion ,Anisotropy ,Mathematics - Abstract
We consider the problem of finding the transversely isotropic elasticity tensor closest to a given elasticity tensor with respect to the Frobenius norm. A similar problem was considered by other authors and solved analytically assuming a fixed orientation of the natural coordinate system of the transversely isotropic tensor. In this paper we formulate a method for finding the optimal orientation of the coordinate system—the one that produces the shortest distance. The optimization problem reduces to finding the absolute maximum of a homogeneous eighth-degree polynomial on a two-dimensional sphere. This formulation allows us a convenient visualization of local extrema, and enables us to find the closest transversely isotropic tensor numerically.
- Published
- 2008
29. Group gradings on simple Lie algebras in positive characteristic
- Author
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Susan Montgomery, Yuri Bahturin, and Mikhail Kochetov
- Subjects
Combinatorics ,Discrete mathematics ,Finite group ,Simple (abstract algebra) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Lie algebra ,Lie group ,Group algebra ,Algebraically closed field ,Abelian group ,Mathematics - Abstract
In this paper we describe all gradings by a finite abelian group G G on the following Lie algebras over an algebraically closed field F F of characteristic p ≠ 2 p\neq 2 : s l n ( F ) \mathfrak {sl}_n(F) ( n n not divisible by p p ), s o n ( F ) \mathfrak {so}_n(F) ( n ≥ 5 n\geq 5 , n ≠ 8 n\neq 8 ) and s p n ( F ) \mathfrak {sp}_n(F) ( n ≥ 6 n\geq 6 , n n even).
- Published
- 2008
30. Engel’s Theorem for Generalized Lie Algebras
- Author
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Mikhail Kochetov and Oana G. Radu
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Engel's theorem ,Killing form ,Affine Lie algebra ,Graded Lie algebra ,Lie conformal algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Lie algebra ,Mathematics - Abstract
We prove analogues of the classical Engel’s Theorem for Lie algebras in the category of comodules over a cotriangular Hopf algebra, generalizing the known result for Lie coloralgebras.
- Published
- 2008
31. Orderings and *-Orderings on Cocommutative Hopf Algebras
- Author
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Jakob Cimprič, Mikhail Kochetov, and Murray Marshall
- Subjects
Discrete mathematics ,Nilpotent ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Hopf algebra ,Noncommutative geometry ,Mathematics ,Group ring - Abstract
Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings \(\Bbbk\)G and smash products U(L)\(\# _{\varphi } \Bbbk\)G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for \(\Bbbk = \mathbb{R}\) and L finite–dimensional, we characterize the orderability of U(L)\(\# _{\varphi } \Bbbk\)G; for \(\Bbbk = \mathbb{C}\), we give a necessary and a sufficient condition for ∗-orderability of \(\Bbbk\)G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for \(\Bbbk = \mathbb{C}\) and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L)\(\# _{\varphi } \Bbbk\)G to the problem of characterizing the ∗-orderability of \(\Bbbk\)G. The latter remains open.
- Published
- 2006
32. PI Hopf algebras of prime characteristic
- Author
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Mikhail Kochetov
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Quantum group ,Mathematics::Rings and Algebras ,010102 general mathematics ,Subalgebra ,Non-associative algebra ,Universal enveloping algebra ,Representation theory of Hopf algebras ,010103 numerical & computational mathematics ,Hopf algebra ,01 natural sciences ,Interior algebra ,Mathematics::Quantum Algebra ,Division algebra ,0101 mathematics ,Mathematics - Abstract
We study the conditions when a cocommutative Hopf algebra of prime characteristic has a polynomial identity (as an algebra). In the case of characteristic 0, this question reduces to smash products of group algebras and universal enveloping algebras, so the answer is known. In characteristic p, however, there are connected Hopf algebras other than restricted envelopes. We find necessary and sufficient conditions of being PI for some extensive classes of such algebras as well as their smash products with group algebras.
- Published
- 2003
33. Polycharacters of Cocommutative Hopf Algebras
- Author
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Yuri Bahturin, Mikhail Kochetov, and Susan Montgomery
- Subjects
Quantum group ,General Mathematics ,010102 general mathematics ,Non-associative algebra ,Universal enveloping algebra ,Representation theory of Hopf algebras ,010103 numerical & computational mathematics ,Hopf algebra ,01 natural sciences ,Algebra ,Interior algebra ,Mathematics::Quantum Algebra ,Division algebra ,Nest algebra ,0101 mathematics ,Mathematics - Abstract
In this paper we extend a well-known theorem of M. Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.
- Published
- 2002
34. On rigidity of Nichols algebras
- Author
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Iván Angiono, Mitja Mastnak, and Mikhail Kochetov
- Subjects
Pure mathematics ,Braided tensor categories ,Matemáticas ,Non-associative algebra ,Rigidity (psychology) ,01 natural sciences ,Matemática Pura ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,Mathematics ,Algebra and Number Theory ,010102 general mathematics ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Hopf algebra ,Algebra ,Bracket (mathematics) ,Hopf algebras ,Rings and Algebras (math.RA) ,17B37 (Primary) 16E40, 16S80, 16T20 (Secondary) ,010307 mathematical physics ,CIENCIAS NATURALES Y EXACTAS - Abstract
We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch-Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of "braided Lie algebras" with nontrivial bracket., Comment: 22 pages
- Published
- 2014
- Full Text
- View/download PDF
35. Gradings on the Lie algebra $D_4$ revisited
- Author
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Alberto Elduque and Mikhail Kochetov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,010102 general mathematics ,Mathematics::Rings and Algebras ,Universal enveloping algebra ,Lie superalgebra ,0102 computer and information sciences ,Mathematics - Rings and Algebras ,01 natural sciences ,Lie conformal algebra ,Graded Lie algebra ,Filtered algebra ,010201 computation theory & mathematics ,Rings and Algebras (math.RA) ,Differential graded algebra ,17B70 (Primary), 17B25, 17C40, 17A75 (Secondary) ,Lie algebra ,Division algebra ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
We classify group gradings on the simple Lie algebra $L$ of type $D_4$ over an algebraically closed field of characteristic different from 2: fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism. For each $G$-grading on $L$, we also study graded $L$-modules (assuming characteristic 0)., Comment: 29 pages
- Published
- 2014
- Full Text
- View/download PDF
36. Composition algebras and type 𝐺₂
- Author
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Mikhail Kochetov and Alberto Elduque
- Subjects
Crystallography ,Chemistry ,Composition (combinatorics) ,Type (model theory) - Published
- 2013
37. Introduction
- Author
-
Alberto Elduque and Mikhail Kochetov
- Published
- 2013
38. Gradings on algebras
- Author
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Alberto Elduque and Mikhail Kochetov
- Published
- 2013
39. Other simple Lie algebras in characteristic zero
- Author
-
Mikhail Kochetov and Alberto Elduque
- Subjects
Discrete mathematics ,Adjoint representation of a Lie algebra ,Pure mathematics ,Representation of a Lie group ,Simple Lie group ,Non-associative algebra ,Killing form ,Affine Lie algebra ,Generalized Kac–Moody algebra ,Lie conformal algebra ,Mathematics - Published
- 2013
40. Lie algebras of Cartan type in prime characteristic
- Author
-
Alberto Elduque and Mikhail Kochetov
- Subjects
Algebra ,Adjoint representation of a Lie algebra ,Pure mathematics ,Cartan decomposition ,Cartan matrix ,Real form ,Cartan subalgebra ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Mathematics - Published
- 2013
41. Affine group schemes
- Author
-
Mikhail Kochetov and Alberto Elduque
- Subjects
Pure mathematics ,Affine group ,Mathematics - Published
- 2013
42. Gradings on Simple Lie Algebras
- Author
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Alberto Elduque and Mikhail Kochetov
- Published
- 2013
43. Classical Lie algebras
- Author
-
Alberto Elduque and Mikhail Kochetov
- Subjects
Pure mathematics ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Non-associative algebra ,Fundamental representation ,Killing form ,Affine Lie algebra ,Lie conformal algebra ,Mathematics ,Graded Lie algebra - Published
- 2013
44. Jordan algebras and type 𝐹₄
- Author
-
Mikhail Kochetov and Alberto Elduque
- Subjects
Algebra ,Jordan algebra ,Type (model theory) ,Mathematics - Published
- 2013
45. Irreducible root systems
- Author
-
Alberto Elduque and Mikhail Kochetov
- Subjects
Pure mathematics ,Root system ,Mathematics - Published
- 2013
46. Counting fine gradings on matrix algebras and on classical simple Lie algebras
- Author
-
Nicholas Parsons, Sergey Sadov, and Mikhail Kochetov
- Subjects
Pure mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Graded ring ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Type (model theory) ,16. Peace & justice ,01 natural sciences ,17B70, 16W50 (Primary) 65A05 (Secondary) ,Matrix (mathematics) ,Simple (abstract algebra) ,Rings and Algebras (math.RA) ,Lie algebra ,FOS: Mathematics ,0101 mathematics ,Algebraically closed field ,Constant (mathematics) ,Mathematics - Abstract
Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $\mathbb{F}$ (assuming $\mathrm{char} \mathbb{F}\ne 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $X\in\{A,C,D\}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r$ with $r\le 100$ as well as the asymptotic behaviour of the average, $\hat N_X(r)$, for large $r$. In particular, we prove that there exist positive constants $b$ and $c$ such that $\exp(br^{2/3})\le\hat N_X(r)\le\exp(cr^{2/3})$. The analogous average for matrix algebras $M_n(\mathbb{F})$ is proved to be $a\ln n+O(1)$ where $a$ is an explicit constant depending on $\mathrm{char} \mathbb{F}$., Comment: 24 pages with 8 tables
- Published
- 2012
- Full Text
- View/download PDF
47. Group gradings on finitary simple Lie algebras
- Author
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Matej Brešar, Mikhail Kochetov, and Yuri Bahturin
- Subjects
Pure mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,01 natural sciences ,Rings and Algebras (math.RA) ,Lie algebra ,FOS: Mathematics ,Finitary ,17B70 (Primary) 16R60, 16W50, 17B60 (Secondary) ,Isomorphism ,0101 mathematics ,Abelian group ,Algebraically closed field ,Symplectic geometry ,Mathematics ,Vector space - Abstract
We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2., Comment: Several typographical errors have been corrected
- Published
- 2011
- Full Text
- View/download PDF
48. Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra
- Author
-
Mikhail Kochetov and Alberto Elduque
- Subjects
Fine grading ,Pure mathematics ,Albert algebra ,Octonions ,Non-associative algebra ,Universal enveloping algebra ,010103 numerical & computational mathematics ,01 natural sciences ,Graded algebra ,symbols.namesake ,Mathematics::Group Theory ,Weyl group ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Algebra and Number Theory ,Jordan algebra ,010102 general mathematics ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Affine Lie algebra ,Rings and Algebras (math.RA) ,symbols ,Cellular algebra ,Generalized Kac–Moody algebra ,Primary 16W50, secondary 17D05, 17C40 - Abstract
Given a grading $\Gamma: A=\oplus_{g\in G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $\Gamma$ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra)., Comment: 23 pages. References updated
- Published
- 2010
- Full Text
- View/download PDF
49. Polynomial Identities in Hopf Algebras: Passman’s Theorem and Its Dual
- Author
-
Mikhail Kochetov
- Subjects
Algebra ,Factor theorem ,Polynomial ,Group (mathematics) ,Mathematics::Quantum Algebra ,Coalgebra ,Algebraic group ,Mathematics::Rings and Algebras ,Group algebra ,Hopf algebra ,Commutative property ,Mathematics - Abstract
We give necessary and sufficient conditions when a cocommutative Hopf algebra of characteristic 0 satisfies an identity as an algebra, which can be considered as a generalization of Passman’s theorem on group algebras. Then we look at the dual situation: when a commutative Hopf algebra (not necessarily of characteristic 0) satisfies an identity as a coalgebra.
- Published
- 2003
50. Engel’s Theorem for Generalized Lie Algebras.
- Author
-
Mikhail Kochetov and Oana Radu
- Abstract
Abstract We prove analogues of the classical Engel’s Theorem for Lie algebras in the category of comodules over a cotriangular Hopf algebra, generalizing the known result for Lie coloralgebras. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
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