Persons
doc. RNDr. Natalie Žukovec, Ph.D.
All publications
THE FREE ALTERNATIVE NIL-SUPERALGEBRA OF INDEX N ON ONE ODD GENERATOR
- Authors: Scholtzová, J., doc. RNDr. Natalie Žukovec, Ph.D.,
- Publication: Communications in Algebra. 2014, 42(1), 96-107. ISSN 0092-7872.
- Year: 2014
- DOI: 10.1080/00927872.2012.706844
- Link: https://doi.org/10.1080/00927872.2012.706844
- Department: Department of Mathematics
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Annotation:
A base of the free alternative nil-superalgebra of index n, for n3, on one odd generator is constructed. In particular, its index of solvability is computed. As a corollary over a field of characteristic zero an infinite family of solvable alternative algebras which are not associative of arbitrary big solvability index with explicit multiplication table is obtained.
CENTRAL AND NUCLEAR SKEW-SYMMETRIC ELEMENTS IN ALTERNATIVE ALGEBRAS
- Authors: doc. RNDr. Natalie Žukovec, Ph.D.,
- Publication: INTERNATIONAL ALGEBRAIC CONFERENCE DEDICATED TO THE 70th BIRTHDAY OF A. V. YAKOVLEV. St. Petersburg: Saint-Petersburg State University, 2010. pp. 168-171.
- Year: 2010
- Department: Department of Mathematics
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Annotation:
Some results on central and nuclear skew-symmetric elements in alternative algebras are presented.
Obtained results about the free alternative nil-superalgebras on one odd generator of different indices
- Authors: Scholtzová, J., doc. RNDr. Natalie Žukovec, Ph.D.,
- Publication: Workshop 2010. Praha: České vysoké učení technické v Praze, 2010. pp. 10-11. CTU Reports. ISBN 978-80-01-04513-8.
- Year: 2010
- Department: Department of Mathematics
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Annotation:
As mentioned in previous articles, we try to obtain some general results in problematic of finding bases and the indices of solvability of alternative nil-algebras of different indices using the superalgebra method. This way, we constructed the bases of the free alternative nil-superalgebras on one odd generator of different indices, more precisely for the nil index 2 and 3 respectively (published in [1,3]). Finding of the base of the alternative nil-superalgebra of nil index 4 and 5, is in process.
THE FREE ALTERNATIVE NIL-SUPERALGEBRA OF INDEX 3 ON ONE ODD GENERATOR
- Authors: Scholtzová, J., doc. RNDr. Natalie Žukovec, Ph.D.,
- Publication: Journal of Algebra and Its Applications (JAA). 2010, 9(2), 209-222. ISSN 0219-4988.
- Year: 2010
- DOI: 10.1142/S0219498810003847
- Link: https://doi.org/10.1142/S0219498810003847
- Department: Department of Mathematics
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Annotation:
A base of the free alternative nil-superalgebra of index 3 on one odd generator is constructed. In particular, its index of solvability is computed. We consider also the corresponding Grassmann algebra and show that the well-known Dorofeev's example of solvable non-nilpotent alternative algebra is its homomorphic image
Alternative Superalgebras on one odd Generator
- Authors: doc. RNDr. Natalie Žukovec, Ph.D.,
- Publication: Diskretnaja matematika, algebra i ich priloženija. Minsk: Institut matematiki Nacionalnoj akademiji nauk Belorusi, 2009. pp. 70-71. ISBN 985-6499-61-5.
- Year: 2009
- Department: Department of Mathematics
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Annotation:
It is a short survey of obtained results on alternative superalgebras with one odd generator.
Skew-symmetric Identities of Octonions
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: Journal of Pure and Applied Algebra. 2009, 213(4), 479-492. ISSN 0022-4049.
- Year: 2009
- DOI: 10.1016/j.jpaa.2008.07.012
- Link: https://doi.org/10.1016/j.jpaa.2008.07.012
- Department: Department of Mathematics
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Annotation:
Quadratic alternative superalgebras are introduced and their super- identities and central functions on one odd generator are described. As a corollary, all multilinear skew-symmetric identities and central polynomials of octonions are classied.
A Base of the Free Alternative Superalgebra on One Odd Generator
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.P.
- Publication: DOKLADY MATHEMATICS. 2008, 78(2), 693-695. ISSN 1064-5624.
- Year: 2008
- DOI: 10.1134/S106456240805013X
- Link: https://doi.org/10.1134/S106456240805013X
- Department: Department of Mathematics
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Annotation:
A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.
The free alternative superalgebra on one odd generator
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: International Journal of Algebra and Computation. 2007, 17(5-6), 1215-1247. ISSN 0218-1967.
- Year: 2007
- Department: Department of Mathematics
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Annotation:
A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.
The free Malcev superalgebra on one odd generator and related superalgebras
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: International Journal of Mathematics and Mathematical Sciences. 2007, 140(2), 243-249. ISSN 0161-1712.
- Year: 2007
- DOI: 10.1007/s10958-007-0421-x
- Link: https://doi.org/10.1007/s10958-007-0421-x
- Department: Department of Mathematics
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Annotation:
A base of the universal multiplicative envelope of free Malcev superalgebra M on one odd generator is constructed. Some corollaries for skew-symmetric functions and central elements in free Malcev and free alternative algebras are obtained. Moreover, a base of the Poisson Malcev superalgebra of M is constructed. As a corollary, a set of elements that spans the free alternative superalgebra on one odd generator is obtained.
Speciality of Malcev superalgebras on one odd generator
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: Journal of Algebra. 2006, 301(2), 587-600. ISSN 0021-8693.
- Year: 2006
- DOI: 10.1016/j.jalgebra.2005.11.003
- Link: https://doi.org/10.1016/j.jalgebra.2005.11.003
- Department: Department of Mathematics
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Annotation:
It is proved that every Malcev superalgebra generated by an odd element is special, that is, isomorphic to a subsuperalgebra of the commutator Malcev superalgebra $A^-$ for a certain alternative superalgebra $A$. As a corollary, it is shown that the kernel of the natural homomorphism of the free Malcev algebra $Malc[X]$ of countable rank into the commutator Malcev algebra $Alt[X]^-$ of the corresponding free alternative algebra $Alt[X]$, does not contain skew-symmetric multilinear elements. In other words, there are no skew-symmetric Malcev $s$-identities. Another corollary is speciality of the Malcev Grassmann algebra.
The Malcev Poisson Superalgebra of the Free Malcev Superalgebra on One Odd Generator
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: Journal of Algebra and Its Applications (JAA). 2006, 5(4), 521-535. ISSN 0219-4988.
- Year: 2006
- Department: Department of Mathematics
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Annotation:
A base of the Malcev Poisson superalgebra related with the free Malcev superalgebra on one odd generator is constructed. As a corollary, a pre-base, that is, a set of elements that spans the free alternative superalgebra on one odd generator is obtained.
The universal multiplicative envelope of the free Malcev superalgebra on one odd generator
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: Communications in Algebra. 2006, 34(4), 1319-1344. ISSN 0092-7872.
- Year: 2006
- Department: Department of Mathematics
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Annotation:
A base of the universal multiplicative envelope for the free Malcev superalgebra on one odd generator is constructed. Some corollaries for central skew-symmetric elements in free Malcev and free alternative algebras are obtained. In particular, the Malcev Grassmann algebra is constructed.
The Neighbourhood of Dihedral 2-groups
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Bálek, M., Drápal, A.
- Publication: Acta Applicandae Mathematicae. 2005, 2005(85), 25-33. ISSN 0167-8019.
- Year: 2005
- DOI: 10.1007/s10440-004-5582-8
- Link: https://doi.org/10.1007/s10440-004-5582-8
- Department: Department of Mathematics
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Annotation:
We examine two particular constructions that derive from a \mbox{$2$-group} $G=G(\cdot)$ another $2$-group $G(*)$ for the case when $G(\cdot)$ is one of $D_{2^n}$, $SD_{2^n}$, $Q_{2^n}$. The constructions (the cyclic one and the dihedral one) have the property that $x*y=x\cdot y$ for exactly $3/4$ of all pairs $(x,y)\in G\times G$.
Free Malcev superalgebra on one odd generator and related superalgebras
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: Fundamental and Applied Mathematics. 2004, 10(4), 97-106. ISSN 1560-5159.
- Year: 2004
- Department: Department of Mathematics
On Associative Algebras Satisfying the Identity $x^5=0$
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Shestakov, I.
- Publication: Algebra and Discrete Mathematics. 2004, 2004(1), 112-120. ISSN 1726-3255.
- Year: 2004
- Department: Department of Mathematics
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Annotation:
We study Kuzmin's conjecture on the index of nilpotency for the variety $\M_5$ of associative nil-algebras of degree 5. Due to Vaughan-Lee~\cite{vau2} the problem is reduced to that for $k$-generator $\M_5$-superalgeb\-ras, where $k\leq 5$. We confirm Kuzmin's conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15
On Finite Degree Partial Representations of Groups
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Dokuchaev, M.
- Publication: Journal of Algebra. 2004, 2004(274), 309-334. ISSN 0021-8693.
- Year: 2004
- DOI: 10.1016/S0021-8693(03)00533-7
- Link: https://doi.org/10.1016/S0021-8693(03)00533-7
- Department: Department of Mathematics
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Annotation:
We establish a one-to-one correspondence between the irreducible finite degree partial representations of a group $G$ and the (usual) irreducible representations of certain ideals of a groupoid algebra constructed from $G$. We derive a structural result about the irreducible partial representations on finite dimensional vector spaces and give the description ``up to usual representations'' of the irreducible partial representations of abelian groups of degrees $\leq4$. We treat simultaneously irreducible and indecomposable partial representations
On multiplication tables of groups that agree on half of the columns and half of the rows
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Drapal, A.
- Publication: Glasgow Mathematical Journal. 2003, 2003(45), 293-308. ISSN 0017-0895.
- Year: 2003
- DOI: 10.1017/S0017089503001253
- Link: https://doi.org/10.1017/S0017089503001253
- Department: Department of Mathematics
On Multiplication Tables of Groups that Agree on Half of the Columns and Half of the Rows
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Drápal, A.
- Publication: Glasgow Mathematical Journal. 2003,(45), 293-308. ISSN 0017-0895.
- Year: 2003
- DOI: 10.1017/S0017089503001253
- Link: https://doi.org/10.1017/S0017089503001253
- Department: Department of Mathematics
On Irreducible Partial Representations of Groups
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Dokuchaev, M.
- Publication: Comptes Rendus Mathematiques de'l Academie des Sciences - Mathematical Reports of the Academy of Science. 2002,(24), 85-90. ISSN 0706-1994.
- Year: 2002
- Department: Department of Mathematics
Partial Actions, Partial Group Rings and Crossed Products
- Authors: doc. RNDr. Natalie Žukovec, Ph.D., Dokuchaev, M.
- Publication: Algebraic Structures and their Applications. Kyjev: Inst. Math. Ukr. Nat. Academy Sci, 2002. pp. 212-223.
- Year: 2002
- Department: Department of Mathematics
On Small Distances of Small 2-groups
- Authors: doc. RNDr. Natalie Žukovec, Ph.D.,
- Publication: Commentationes Mathematicae Universitatis Carolinae. 2001, 42(2), 247-257. ISSN 0010-2628.
- Year: 2001
- Department: Department of Mathematics