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Data publikacji: 4 czerwca 2021

Data aktualizacji: 30 lipca 2024

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1

KORNYAK, V. V. "COMPUTATION OF COHOMOLOGY OF LIE SUPERALGEBRAS OF VECTOR FIELDS". International Journal of Modern Physics C 11, nr 02 (marzec 2000): 397–413. http://dx.doi.org/10.1142/s0129183100000353.

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The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras, Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.

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2

Dadarlat, Marius. "Fiberwise KK-equivalence of continuous fields of C*-algebras". Journal of K-Theory 3, nr 2 (28.05.2008): 205–19. http://dx.doi.org/10.1017/is008001012jkt041.

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AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

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3

Iovanov, Miodrag Cristian, i Alexander Harris Sistko. "Maximal subalgebras of finite-dimensional algebras". Forum Mathematicum 31, nr 5 (1.09.2019): 1283–304. http://dx.doi.org/10.1515/forum-2019-0033.

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AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

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4

Mounirh, Karim. "Nicely semiramified division algebras over Henselian fields". International Journal of Mathematics and Mathematical Sciences 2005, nr 4 (2005): 571–77. http://dx.doi.org/10.1155/ijmms.2005.571.

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This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

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5

Fratila, Dragos. "Cusp eigenforms and the hall algebra of an elliptic curve". Compositio Mathematica 149, nr 6 (4.03.2013): 914–58. http://dx.doi.org/10.1112/s0010437x12000784.

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AbstractWe give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

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6

GORAZD, TOMASZ A. "FAST ISOMORPHISM TESTING IN ARITHMETICAL VARIETIES". International Journal of Algebra and Computation 13, nr 04 (sierpień 2003): 499–506. http://dx.doi.org/10.1142/s0218196703001572.

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Let [Formula: see text] be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from [Formula: see text] have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from [Formula: see text]. This includes the following classical structures in algebra: • Boolean algebras. • Varieties of rings generated by finitely many finite fields. • Varieties of Heyting algebras generated by an n–element chain.

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7

BILLIG, YULY. "MODULES FOR A SHEAF OF LIE ALGEBRAS ON LOOP MANIFOLDS". International Journal of Mathematics 23, nr 08 (10.07.2012): 1250079. http://dx.doi.org/10.1142/s0129167x12500796.

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We consider a semidirect product of the sheaf of vector fields on a manifold ℂ* × X with a central extension of the sheaf of Lie algebras of maps from ℂ* × X into a finite-dimensional simple Lie algebra, viewed as sheaves on X. Using vertex algebra methods we construct sheaves of modules for this sheaf of Lie algebras. Our results extend the work of Malikov–Schechtman–Vaintrob on the chiral de Rham complex.

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8

MAYR, PETER. "THE SUBPOWER MEMBERSHIP PROBLEM FOR MAL'CEV ALGEBRAS". International Journal of Algebra and Computation 22, nr 07 (listopad 2012): 1250075. http://dx.doi.org/10.1142/s0218196712500750.

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Given tuples a1, …, ak and b in An for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of An that is generated by a1, …, ak. For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal'cev algebra is in NP and give a polynomial time algorithm for any finite Mal'cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.

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9

Regev, Amitai. "Grassmann algebras over finite fields". Communications in Algebra 19, nr 6 (styczeń 1991): 1829–49. http://dx.doi.org/10.1080/00927879108824231.

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10

PÉREZ, EFRÉN. "ON SEMIGENERIC TAMENESS AND BASE FIELD EXTENSION". Glasgow Mathematical Journal 58, nr 1 (21.07.2015): 39–53. http://dx.doi.org/10.1017/s0017089515000051.

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AbstractThe notions of central endolength and semigeneric tameness are introduced, and their behaviour under base field extension for finite-dimensional algebras over perfect fields are analysed. Forka perfect field,Kan algebraic closure and Λ a finite-dimensionalk-algebra, here there is a proof that Λ is semigenerically tame if and only if Λ ⊗kKis tame.

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11

Izumi, M., i T. Sogabe. "The group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra". International Journal of Mathematics 30, nr 11 (październik 2019): 1950057. http://dx.doi.org/10.1142/s0129167x19500575.

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We determine the group structure of the homotopy set whose target is the automorphism group of the Cuntz algebra [Formula: see text] for finite [Formula: see text] in terms of K-theory. We show that there is an example of a space for which the homotopy set is a noncommutative group, and hence, the classifying space of the automorphism group of the Cuntz algebra for finite [Formula: see text] is not an H-space. We also make an improvement of Dadarlat’s classification of continuous fields of the Cuntz algebras in terms of vector bundles.

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12

Dräxler, Peter. "Representation-Directed Diamonds". LMS Journal of Computation and Mathematics 4 (2001): 14–21. http://dx.doi.org/10.1112/s1461157000000784.

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AbstractA module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

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13

ABRATE, MARCO. "QUADRATIC FORMULAS FOR GENERALIZED QUATERNIONS". Journal of Algebra and Its Applications 08, nr 03 (czerwiec 2009): 289–306. http://dx.doi.org/10.1142/s0219498809003308.

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In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.

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14

Ghaani Farashahi, Arash. "Wave Packet Transform over Finite Fields". Electronic Journal of Linear Algebra 30 (8.02.2015): 507–29. http://dx.doi.org/10.13001/1081-3810.2903.

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In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.

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15

Herman, Allen. "A Constructive Brauer-Witt Theorem for Certain Solvable Groups". Canadian Journal of Mathematics 48, nr 6 (1.12.1996): 1196–209. http://dx.doi.org/10.4153/cjm-1996-063-1.

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AbstractDivision algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

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16

Fitzgerald, Robert W., i Yasanthi Kottegoda. "Power trace functions over finite fields". Journal of Algebra and Its Applications 19, nr 10 (4.10.2019): 2050196. http://dx.doi.org/10.1142/s0219498820501960.

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We count the number of solutions to a power trace function equal to a constant and use this to find the probability of a successful attack on an authentication code proposed by Ding et al. (2005) [C. Ding, A. Salomaa, P. Solé and X. Tian, Three constructions of authentication/secrecy codes, J. Pure Appl. Algebra 196 (2005) 149–168].

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17

Falcón, Óscar J., Raúl M. Falcón, Juan Núñez, Ana M. Pacheco i M. Trinidad Villar. "Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields". Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, nr 2 (1.06.2016): 185–204. http://dx.doi.org/10.1515/auom-2016-0036.

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Abstract This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomorphism, six, five and five 6-dimensional filiform Lie algebras and fifteen, eleven and fifteen 7-dimensional ones, respectively, over ℤ/pℤ, for p = 2, 3, 5. In any case, the main interest of the paper is not the computations itself but both to provide new strategies to find out properties of Lie algebras and to exemplify a suitable technique to be used in classifications for larger dimensions.

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18

Lenstra, H. W. "Automorphisms of finite fields". Journal of Number Theory 34, nr 1 (styczeń 1990): 33–40. http://dx.doi.org/10.1016/0022-314x(90)90050-2.

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19

Bani-Ata, Mashhour, i Mariam Al-Rashed. "On certain finite dimensional algebras over finite fields". Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 58, nr 1 (19.08.2016): 195–200. http://dx.doi.org/10.1007/s13366-016-0312-8.

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20

SÁNCHEZ, OMAR LEÓN, i RAHIM MOOSA. "THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS". Journal of Symbolic Logic 81, nr 2 (czerwiec 2016): 493–509. http://dx.doi.org/10.1017/jsl.2015.76.

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AbstractA model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

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21

LOPATIN, ARTEM A., i IVAN P. SHESTAKOV. "ASSOCIATIVE NIL-ALGEBRAS OVER FINITE FIELDS". International Journal of Algebra and Computation 23, nr 08 (grudzień 2013): 1881–94. http://dx.doi.org/10.1142/s0218196713500471.

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We study the nilpotency degree of a relatively free finitely generated associative algebra with the identity xn = 0 over a finite field 𝔽 with q elements. In the case of q ≥ n the nilpotency degree is proven to be the same as in the case of an infinite field of the same characteristic. In the case of q = n - 1 it is shown that the nilpotency degree differs from the nilpotency degree for an infinite field of the same characteristic by at most one. The nilpotency degree is explicitly computed for n = 3.

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22

Farkas, Daniel R. "Birational invariants of crystals and fields with a finite group of operators". Mathematical Proceedings of the Cambridge Philosophical Society 107, nr 3 (maj 1990): 417–24. http://dx.doi.org/10.1017/s0305004100068717.

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It is well known that an n-dimensional crystallographic group can be reconstructed from its point group, the integral representation of the point group which arises from its action on the translation lattice, and the 2-cocycle which glues the point group to the lattice ([2]). In practice, this constitutes a complicated list of invariants. When confronted with the classification of objects possessing a rich structure, the algebraic geometer first attempts to find more coarse birational invariants. We begin such a programme for torsion-free crystallographic groups. More precisely, if Γ is a torsion-free crystallographic group and k is a field then the group algebra k[Γ] is a non-commutative domain (see [6], chapter 13). It can be localized at its centre to yield a division algebra k(Γ) which is a crossed product; the Galois group is the point group and it acts on the rational function field generated by k and the lattice (regarded multiplicatively), which is a maximal subfield ([3]). What are thecommon invariants of Γ1 and Γ2 when k(Γ1) and k(Γ2) are isomorphic k-algebras?

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23

Hua, Jiuzhao. "Polynomial equations for matrices over finite fields". Bulletin of the Australian Mathematical Society 59, nr 1 (luty 1999): 59–64. http://dx.doi.org/10.1017/s0004972700032603.

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Let E(x) be a monic polynomial over the finite field q of q elements. A formula for the number of n × n matrices θ over q, satisfying E(θ) = 0 is obtained by counting the representations of the algebra q[x]/(E(x)) of degree n. This simplifies a formula of Hodges.

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24

MAGNANO, G., i F. MAGRI. "POISSON-NIJENHUIS STRUCTURES AND SATO HIERARCHY". Reviews in Mathematical Physics 03, nr 04 (grudzień 1991): 403–66. http://dx.doi.org/10.1142/s0129055x91000151.

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We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

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25

Anglès, B., F. Pellarin, F. Tavares Ribeiro i F. Demeslay. "Arithmetic of positive characteristic -series values in Tate algebras". Compositio Mathematica 152, nr 1 (7.09.2015): 1–61. http://dx.doi.org/10.1112/s0010437x15007563.

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The second author has recently introduced a new class of$L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these$L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.

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26

Sultanov, A. Ya, M. V. Glebova i O. V. Bolotnikova. "Lie algebras of differentiations of linear algebras over a field". Differential Geometry of Manifolds of Figures, nr 52 (2021): 123–36. http://dx.doi.org/10.5922/0321-4796-2021-52-12.

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In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equations. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of applying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an example, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.

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27

Glasby, S. P., Frederico A. M. Ribeiro i Csaba Schneider. "Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, nr 4 (25.02.2019): 1827–52. http://dx.doi.org/10.1017/prm.2018.159.

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AbstractLet p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative 𝔽p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, 𝔽).

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28

Kemper, Gregor, Artem Lopatin i Fabian Reimers. "Separating invariants over finite fields". Journal of Pure and Applied Algebra 226, nr 4 (kwiecień 2022): 106904. http://dx.doi.org/10.1016/j.jpaa.2021.106904.

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29

Keel, Sean. "Polarized Pushouts over Finite Fields". Communications in Algebra 31, nr 8 (9.01.2003): 3955–82. http://dx.doi.org/10.1081/agb-120022449.

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30

Akbik, Safwan. "Normal generators of finite fields". Journal of Number Theory 41, nr 2 (czerwiec 1992): 146–49. http://dx.doi.org/10.1016/0022-314x(92)90114-5.

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31

Garcia, A., i J. F. Voloch. "Fermat curves over finite fields". Journal of Number Theory 30, nr 3 (listopad 1988): 345–56. http://dx.doi.org/10.1016/0022-314x(88)90007-8.

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32

Biswas, Indranil, i S. Subramanian. "Principal Bundles Over Finite Fields". Communications in Algebra 39, nr 7 (lipiec 2011): 2268–75. http://dx.doi.org/10.1080/00927872.2010.480962.

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33

Almotairi, Eman S., Rania Kammoun i Ahmad M. Alghamdi. "GRAPH THEORY FOR FINITE FIELDS". JP Journal of Algebra, Number Theory and Applications 63, nr 2 (12.02.2024): 131–51. http://dx.doi.org/10.17654/0972555524008.

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34

VERSTEGEN, DIRK. "ON THE CLASSIFICATION OF W-ALGEBRAS". International Journal of Modern Physics A 10, nr 10 (20.04.1995): 1413–48. http://dx.doi.org/10.1142/s0217751x95000681.

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We review and extend the conformal bootstrap approach to the classification of quantum W-algebras. These are extensions of the Virasoro algebra by a finite set of primary fields. Explicit forms are given for the most general crossing-symmetric four-point functions. Together with a large c expansion of the conformal blocks, this gives a powerful tool for finding all W-algebras that are associative for generic values of the central charge c.

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35

Matsuo, Yutaka. "Free fields and quasi-finite representation of 1+∞ algebra". Physics Letters B 326, nr 1-2 (kwiecień 1994): 95–100. http://dx.doi.org/10.1016/0370-2693(94)91198-3.

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36

Ma, Jia-jun, Congling Qiu i Jialiang Zou. "Generic Hecke algebra and theta correspondence over finite fields". Advances in Mathematics 438 (luty 2024): 109444. http://dx.doi.org/10.1016/j.aim.2023.109444.

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37

SCHRAML, STEFAN. "ENVELOPING ALGEBRA-VALUED GAUGE TRANSFORMATIONS ON NONCOMMUTATIVE SPACES". Modern Physics Letters A 16, nr 04n06 (28.02.2001): 337–41. http://dx.doi.org/10.1142/s0217732301003437.

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38

Regev, Amitai. "Remarks on P.I. algebras over finite fields". Journal of Algebra 145, nr 1 (styczeń 1992): 249–61. http://dx.doi.org/10.1016/0021-8693(92)90191-n.

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39

Bolokhov, Sergey V., i Vladimir D. Ivashchuk. "Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras". Symmetry 15, nr 6 (3.06.2023): 1199. http://dx.doi.org/10.3390/sym15061199.

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This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions Hs(z) comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters ps>0, s=1,…,n. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks n=1,2,3,4,5 and exceptional algebra E6. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored.

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40

Tryhuk, V., V. Chrastinová i O. Dlouhý. "The Lie Group in Infinite Dimension". Abstract and Applied Analysis 2011 (2011): 1–35. http://dx.doi.org/10.1155/2011/919538.

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A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

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41

Munn, W. D. "Involutions on finite-dimensional algebras over real closed fields". Journal of the Australian Mathematical Society 77, nr 1 (sierpień 2004): 123–28. http://dx.doi.org/10.1017/s1446788700010193.

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AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

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42

Özbudak, Ferruh, i Michael Thomas. "A note on towers of function fields over finite fields". Communications in Algebra 26, nr 11 (styczeń 1998): 3737–41. http://dx.doi.org/10.1080/00927879808826370.

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43

Ma, Liming, Chaoping Xing i Sze Ling Yeo. "On automorphism groups of cyclotomic function fields over finite fields". Journal of Number Theory 169 (grudzień 2016): 406–19. http://dx.doi.org/10.1016/j.jnt.2016.05.026.

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44

Özbudak, Ferruh, i Michael Thomas. "A note on towers of function fields over finite fields". Communications in Algebra 28, nr 10 (styczeń 2000): 4729–33. http://dx.doi.org/10.1080/00927870008827116.

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45

Baoulina, Ioulia. "On some equations over finite fields". Journal de Théorie des Nombres de Bordeaux 17, nr 1 (2005): 45–50. http://dx.doi.org/10.5802/jtnb.475.

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46

Knudson, Kevin P. "Unstable homotopy invariance for finite fields". Fundamenta Mathematicae 175, nr 2 (2002): 155–62. http://dx.doi.org/10.4064/fm175-2-5.

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47

Avilés, Antonio. "COMMUTATIVE RINGS WITH FINITE QUOTIENT FIELDS". Communications in Algebra 33, nr 3 (9.03.2005): 727–36. http://dx.doi.org/10.1081/agb-200049882.

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48

Brydon, Duncan. "Separable exterior squares over finite fields". Journal of Algebra 268, nr 2 (październik 2003): 700–722. http://dx.doi.org/10.1016/s0021-8693(03)00272-2.

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49

Bornhofen, Matthias, i Urs Hartl. "Pure Anderson motives over finite fields". Journal of Number Theory 129, nr 2 (luty 2009): 247–83. http://dx.doi.org/10.1016/j.jnt.2008.09.006.

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50

Bodine, E. J., i J. J. McDonald. "Spectrally arbitrary patterns over finite fields". Linear and Multilinear Algebra 60, nr 3 (marzec 2012): 285–99. http://dx.doi.org/10.1080/03081087.2011.591395.

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