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Search: MSC category 17 ( Nonassociative rings and algebras )

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1. CMB Online first

Liu, Ling; Makhlouf, Abdenacer; Menini, Claudia; Panaite, Florin
$\{\sigma , \tau \}$-Rota-Baxter operators, infinitesimal Hom-bialgebras and the associative (Bi)Hom-Yang-Baxter equation
We introduce the concept of $\{\sigma , \tau \}$-Rota-Baxter operator, as a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain $\{\sigma , \tau \}$-Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

Keywords:Rota-Baxter operator, Hom-pre-Lie algebra, infinitesimal Hom-bialgebra, associative (Bi)Hom-Yang-Baxter equation.
Categories:15A04, 17A99, 17D99

2. CMB Online first

Criswell, Jackson; Salisbury, Ben; Tingley, Peter W.
PBW bases and marginally large tableaux in types B and C
We explicitly describe the isomorphism between two combinatorial realizations of Kashiwara's infinity crystal in types B and C. The first realization is in terms of marginally large tableaux and the other is in terms of Kostant partitions coming from PBW bases. We also discuss a stack notation for Kostant partitions which simplifies that realization.

Keywords:crystal, Kostant partition, Lusztig data, marginally large tableau
Categories:05E10, 17B37

3. CMB Online first

Leandro, Cagliero; Szechtman, Fernando
Jordan-Chevalley decomposition in Lie algebras
We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.

Keywords:solvable Lie algebra, Jordan-Chevalley decomposition, representation
Categories:17-08, 17B05, 20C40, 15A21

4. CMB Online first

Bavula, V. V.; Lu, T.
The universal enveloping algebra of the Schrödinger algebra and its prime spectrum
The prime, completely prime, maximal and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. For all of these ideals their explicit generators are given. A counterexample is constructed to the conjecture of Cheng and Zhang about non-existence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds 'generically'.

Keywords:prime ideal, weight module, simple module, centralizer, Whittaker module
Categories:17B10, 16D25, 16D60, 16D70, 16P50

5. CMB 2017 (vol 61 pp. 16)

Bavula, V. V.; Lu, T.
Classification of simple weight modules over the Schrödinger algebra
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.

Keywords:weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra
Categories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50

6. CMB 2017 (vol 61 pp. 318)

Lee, Tsiu-Kwen
Ad-nilpotent Elements of Semiprime Rings with Involution
Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\gt 1$ is a positive integer. We characterize $a\in K$, the Lie algebra of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This generalizes both Martindale and Miers' theorem and the theorem of Brox et al. To prove it we first prove that if $a, b\in R$ satisfy $(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on $R$, where either $n$ is even or $b=0$, then $\big(a-\lambda\big)^{[\frac{n+1}{2}]}=0$ for some $\lambda\in C$.

Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan element
Categories:16N60, 16W10, 17B60

7. CMB 2015 (vol 58 pp. 692)

Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.
Sur les algèbres de Lie associées à une connexion
Let $\Gamma$ be a connection on a smooth manifold $M$, in this paper we give some properties of $\Gamma$ by studying the corresponding Lie algebras. In particular, we compute the first Chevalley-Eilenberg cohomology space of the horizontal vector fields Lie algebra on the tangent bundle of $M$, whose the corresponding Lie derivative of $\Gamma$ is null, and of the horizontal nullity curvature space.

Keywords:algèbre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dérivée de Lie correspondante à une connexion est nulle, espace de nullité de la courbure
Categories:17B66, 53B15

8. CMB 2015 (vol 58 pp. 363)

Sharma, R. K.; Sidana, Swati
Finite Semisimple Loop Algebras of Indecomposable $RA$ Loops
There are at the most seven classes of finite indecomposable $RA$ loops upto isomorphism. In this paper, we completely characterize the structure of the unit loop of loop algebras of these seven classes of loops over finite fields of characteristic greater than $2$.

Keywords:unit loop, loop algebra, indecomposable $RA$ loops
Categories:20N05, 17D05

9. CMB 2015 (vol 58 pp. 233)

Bergen, Jeffrey
Affine Actions of $U_q(sl(2))$ on Polynomial Rings
We classify the affine actions of $U_q(sl(2))$ on commutative polynomial rings in $m \ge 1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m-1$ variables, depending upon whether $q$ is or is not a root of $1$.

Keywords:skew derivation, quantum group, invariants
Categories:16T20, 17B37, 20G42

10. CMB 2014 (vol 58 pp. 69)

Fulp, Ronald Owen
Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies"
The Theorem below is a correction to Theorem 3.5 in the article entitled " Infinite Dimensional DeWitt Supergroups and Their Bodies" published in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part (iii) of that Theorem requires correction. The proof of Theorem 3.5 in the original article failed to separate the proof of (ii) from the proof of (iii). The proof of (ii) is complete once it is established that $ad_a$ is quasi-nilpotent for each $a$ since it immediately follows that $K$ is quasi-nilpotent. The proof of (iii) is not complete in the original article. The revision appears as the proof of (iii) of the revised Theorem below.

Keywords:super groups, body of super groups, Banach Lie groups
Categories:58B25, 17B65, 81R10, 57P99

11. CMB 2014 (vol 57 pp. 735)

Cagliero, Leandro; Szechtman, Fernando
On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras
We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$?

Keywords:uniserial module, Lie algebra, associative algebra, primitive element
Categories:17B10, 13C05, 12F10, 12E20

12. CMB 2012 (vol 56 pp. 606)

Mazorchuk, Volodymyr; Zhao, Kaiming
Characterization of Simple Highest Weight Modules
We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent action
Categories:17B20, 17B65, 17B66, 17B68

13. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
Left Invariant Einstein-Randers Metrics on Compact Lie Groups
In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature
Categories:17B20, 22E46, 53C12

14. CMB 2011 (vol 55 pp. 579)

Ndogmo, J. C.
Casimir Operators and Nilpotent Radicals
It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants
Categories:16W25, 17B45, 16S30

15. CMB 2011 (vol 55 pp. 351)

MacDougall, J. A.; Sweet, L. G.
Rational Homogeneous Algebras
An algebra $A$ is homogeneous if the automorphism group of $A$ acts transitively on the one-dimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\operatorname{dim} A>1$, then $A^{2}=0$.

Keywords:non-associative algebra, homogeneous, automorphism
Categories:17D99, 17A36

16. CMB 2011 (vol 55 pp. 67)

Cummins, C. J.; Duncan, J. F.
An $E_8$ Correspondence for Multiplicative Eta-Products
We describe an $E_8$ correspondence for the multiplicative eta-products of weight at least $4$.

Keywords:We describe an E8 correspondence for the multiplicative eta-products of weight at least 4.
Categories:11F20, 11F12, 17B60

17. CMB 2011 (vol 54 pp. 442)

García, Esther; Lozano, Miguel Gómez; Neher, Erhard
Nondegeneracy for Lie Triple Systems and Kantor Pairs
We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic $0$ are nondegenerate.

Keywords:Kantor pairs, Lie triple systems, Lie algebras
Categories:17A40, 17B60, 17B99

18. CMB 2011 (vol 54 pp. 519)

Neeb, K. H.; Penkov, I.
Erratum: Cartan Subalgebras of $\mathrm{gl}_\infty$
We correct an oversight in the the paper Cartan Subalgebras of $\mathrm{gl}_\infty$, Canad. Math. Bull. 46(2003), no. 4, 597-616. doi: 10.4153/CMB-2003-056-1

Categories:17B65, 17B20

19. CMB 2011 (vol 54 pp. 297)

Johnson, Marianne; Stöhr, Ralph
Lie Powers and Pseudo-Idempotents
We give a new factorisation of the classical Dynkin operator, an element of the integral group ring of the symmetric group that facilitates projections of tensor powers onto Lie powers. As an application we show that the iterated Lie power $L_2(L_n)$ is a module direct summand of the Lie power $L_{2n}$ whenever the characteristic of the ground field does not divide $n$. An explicit projection of the latter onto the former is exhibited in this case.

Categories:17B01, 20C30

20. CMB 2010 (vol 54 pp. 44)

Cheung, Wai-Shun; Tam, Tin-Yau
Star-Shapedness and $K$-Orbits in Complex Semisimple Lie Algebras
Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in \mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$, where $f$ is a linear functional on $\mathfrak{g}$. We establish the star-shapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras of type $B$.

Categories:22E10, 17B20

21. CMB 2010 (vol 53 pp. 425)

Chapoton, Frédéric
Free Pre-Lie Algebras are Free as Lie Algebras
We prove that the $\mathfrak{S}$-module $\operatorname{PreLie}$ is a free Lie algebra in the category of $\mathfrak{S}$-modules and can therefore be written as the composition of the $\mathfrak{S}$-module $\operatorname{Lie}$ with a new $\mathfrak{S}$-module $X$. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using $X$. Furthermore, we define a natural filtration on the $\mathfrak{S}$-module $X$. We also obtain a relationship between $X$ and the $\mathfrak{S}$-module coming from the anticyclic structure of the $\operatorname{PreLie}$ operad.

Categories:18D50, 17B01, 18G40, 05C05

22. CMB 2009 (vol 52 pp. 245)

Goodaire, Edgar G.; Milies, César Polcino
Involutions of RA Loops
Let $L$ be an RA loop, that is, a loop whose loop ring over any coefficient ring $R$ is an alternative, but not associative, ring. Let $\ell\mapsto\ell^\theta$ denote an involution on $L$ and extend it linearly to the loop ring $RL$. An element $\alpha\in RL$ is \emph{symmetric} if $\alpha^\theta=\alpha$ and \emph{skew-symmetric} if $\alpha^\theta=-\alpha$. In this paper, we show that there exists an involution making the symmetric elements of $RL$ commute if and only if the characteristic of $R$ is $2$ or $\theta$ is the canonical involution on $L$, and an involution making the skew-symmetric elements of $RL$ commute if and only if the characteristic of $R$ is $2$ or $4$.

Categories:20N05, 17D05

23. CMB 2009 (vol 40 pp. 143)

Bremner, Murray
Quantum deformations of simple Lie algebras
It is shown that every simple complex Lie algebra $\fg$ admits a 1-parameter family $\fg_q$ of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for $U_q(\fg)$-modules; here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$. From this it follows that the multiplication on $\fg_q$ is $U_q(\fg)$-invariant. In the special case $\fg = {\ss}(2)$, the structure constants for the deformation ${\ss}(2)_q$ are obtained from the quantum Clebsch-Gordan formula applied to $V(2)_q \otimes V(2)_q$; here $V(2)_q$ is the simple 3-dimensional $U_q\bigl({\ss}(2)\bigr)$-module of highest weight $q^2$.

Categories:17B37, 17A01

24. CMB 2009 (vol 40 pp. 103)

Riley, David M.; Tasić, Vladimir
The transfer of a commutator law from a nil-ring to its adjoint group
For every field $F$ of characteristic $p\geq 0$, we construct an example of a finite dimensional nilpotent $F$-algebra $R$ whose adjoint group $A(R)$ is not centre-by-metabelian, in spite of the fact that $R$ is Lie centre-by-metabelian and satisfies the identities $x^{2p}=0$ when $p>2$ and $x^8=0$ when $p=2$. The existence of such algebras answers a question raised by A.~E.~Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4.

Categories:16U60, 17B60

25. CMB 2008 (vol 51 pp. 291)

Spinelli, Ernesto
Group Algebras with Minimal Strong Lie Derived Length
Let $KG$ be a non-commutative strongly Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$. In this note we state necessary and sufficient conditions so that the strong Lie derived length of $KG$ assumes its minimal value, namely $\lceil \log_{2}(p+1)\rceil $.

Keywords:group algebras, strong Lie derived length
Categories:16S34, 17B30
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