A note on strongly split Lie algebras
Split Lie algebras are maybe the most known examples of graded Lie algebras. Since an important category in the class of graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split Lie algebras $L$ and show that if $L$ is centerless, th...
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| Дата: | 2018 |
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Institute of Mathematics, NAS of Ukraine
2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507423022776320 |
|---|---|
| author | Calderón, Martín A. J. Кальдерон, Мартін А. Дж. |
| author_facet | Calderón, Martín A. J. Кальдерон, Мартін А. Дж. |
| author_sort | Calderón, Martín A. J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:20:38Z |
| description | Split Lie algebras are maybe the most known examples of graded Lie algebras. Since an important category in the class of
graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split
Lie algebras $L$ and show that if $L$ is centerless, then $L$ is the direct sum of split ideals each of which is a split-simple
strongly split Lie algebra. |
| first_indexed | 2026-03-24T02:09:04Z |
| format | Article |
| fulltext |
UDC 512.5
A. J. Calderón Martı́n (Univ. Cádiz, Spain)
A NOTE ON STRONGLY SPLIT LIE ALGEBRAS*
ЗАУВАЖЕННЯ ЩОДО СИЛЬНО РОЗЩЕПЛЕНИХ АЛГЕБР ЛI
Split Lie algebras are maybe the most known examples of graded Lie algebras. Since an important category in the class of
graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split
Lie algebras \frakL and show that if \frakL is centerless, then \frakL is the direct sum of split ideals each of which is a split-simple
strongly split Lie algebra.
Розщепленi алгебри Лi є мабуть найбiльш вiдомим прикладом градуйованих алгебр Лi. Оскiльки важливою ка-
тегорiєю в класi градуйованих алгебр є категорiя сильно градуйованих алгебр, ми вводимо (природним чином)
категорiю сильно розщеплених алгебр Лi \frakL i доводимо, що у випадку, коли \frakL не має центра, \frakL є прямою сумою
розщеплених iдеалiв, кожний з яких є просто-розщепленою сильно розщепленою алгеброю Лi.
1. Introduction and preliminaries. We begin by noting that, troughout this short note, all of the
Lie algebras are considered of arbitrary dimension and over an arbitrary base field \BbbK .
By the one hand, let us recall that given a Lie algebra L and a fixed maximal Abelian subalgebra
H of L, we can consider for any linear functional \alpha : H - \rightarrow \BbbK , the root space of L associated to \alpha
as the linear subspace
L\alpha = \{ v\alpha \in L : [h, v\alpha ] = \alpha (h)v\alpha for any h \in H\} .
The elements \alpha \in H\ast satisfying L\alpha \not = 0 are called roots of L with respect to H and if we denote
by \Lambda := \{ \alpha \in H\ast \setminus \{ 0\} : L\alpha \not = 0\} , it is said that L is a split Lie algebra (with respect to H ), if
L = H \oplus
\Biggl( \bigoplus
\alpha \in \Lambda
L\alpha
\Biggr)
.
It is also said that \Lambda is the root system of L, being \Lambda called symmetric if \Lambda = - \Lambda .
Let us focuss for a while on the concept of split-ideal in the framework of split Lie algebras.
Observe that the set of linear mappings \{ \mathrm{a}\mathrm{d}(h) : h \in H\} , where \mathrm{a}\mathrm{d}(h) : L \rightarrow L is defined by
\mathrm{a}\mathrm{d}(h)(v) = [h, v], is a commuting set of diagonalizable endomorphisms. Hence, given any ideal I
of L, since I is invariant under this set, we can write
I = (I \cap H)\oplus
\Biggl( \bigoplus
\alpha \in \Lambda
(I \cap L\alpha )
\Biggr)
. (1)
From here, if I \cap H \not = 0, then I adopts a split like expression (respect to I \cap H ). This motivate
us to introduce the concept of split-ideal as follows. An ideal I of a split Lie algebra L is called a
split-ideal if I \cap H \not = 0. The split Lie algebra L will be called split-simple if [L,L] \not = 0 and L has
not proper split-ideals.
* Suppoted by the PCI of the UCA "Teoria de Lie y Teoria de Espasios de Banach by the PAI wiht ptoject numbers
FQM298, FQM7156 and by the project of the Spanish Ministerio de Education y Ciencia MTM2016-76327C31P.
c\bigcirc A. J. CALDERÓN MARTIN, 2018
988 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
A NOTE ON STRONGLY SPLIT LIE ALGEBRAS 989
By the other hand, we also recall that a graded algebra
A =
\bigoplus
g\in G
Ag,
that is, A is the direct sum of linear subspaces indexed by the elements in an Abelian group (G,+)
in such a way that AgAh \subset Ag+h , is called a strongly graded algebra if the condition AgAh = Ag+h
holds for any g, h \in G (see [2, 3]).
Since Jacobi identity shows that in any split Lie algebra L we have [L\alpha , L\beta ] \subset L\alpha +\beta for any
\alpha , \beta \in \Lambda \cup \{ 0\} and the fact H = L0 holds, we have that L becomes a graded Lie algebra by means
of the Abelian free group generated by \Lambda . Taking into account the above observations we introduce
the category of strongly split Lie algebras as follows.
Definition 1.1. A split Lie algebra \frakL with set of nonzero roots \Lambda is called a strongly split
Lie algebra if H =
\sum
\alpha \in \Lambda [\frakL \alpha ,\frakL - \alpha ] and given \alpha , \beta \in \Lambda such that \alpha + \beta \in \Lambda , then we have
[\frakL \alpha ,\frakL \beta ] = \frakL \alpha +\beta .
As examples of strongly split Lie algebra we can consider the finite dimensional semisimple Lie
algebras, the Lie H\ast -algebras, the locally finite split Lie algebras, the split graded Lie algebras with
only integer roots and the split Lie algebras considered in [1] among other classes of Lie algebras
(see [4 – 6]).
2. Main results. In the following, \frakL denotes a strongly split Lie algebra with a symmetric root
system and \frakL = H \oplus (
\bigoplus
\alpha \in \Lambda
\frakL \alpha ) the corresponding root spaces decomposition.
Definition 2.1. Let \alpha \in \Lambda and \beta \in \Lambda be two nonzero roots. We say that \alpha is connected to \beta if
there exists a family \alpha 1, \alpha 2, . . . , \alpha n \in \Lambda satisfying the following conditions:
1) \alpha 1 = \alpha ,
2) \{ \alpha 1 + \alpha 2, \alpha 1 + \alpha 2 + \alpha 3, . . . , \alpha 1 + . . .+ \alpha n - 1\} \subset \Lambda ,
3) \alpha 1 + \alpha 2 + . . .+ \alpha n = \epsilon \beta \beta for some \epsilon \beta \in \{ \pm 1\} .
We also say that \{ \alpha 1, . . . , \alpha n\} is a connection from \alpha to \beta .
It is straightforward to verify the relation connection is an equivalence relation. In particular,
\alpha \sim - \alpha . So we can consider the quotient set \Lambda / \sim = \{ [\alpha ] : \alpha \in \Lambda \} . Now, for any [\alpha ] \in \Lambda / \sim we
are going to consider the linear subspace \frakL [\alpha ] := H[\alpha ] \oplus V[\alpha ], where H[\alpha ] :=
\sum
\beta \in [\alpha ][\frakL \beta ,\frakL - \beta ] \subset H
and V[\alpha ] :=
\bigoplus
\beta \in [\alpha ]
\frakL \beta .
Proposition 2.1. Any \frakL [\alpha ] is an ideal of \frakL . If furthermore \frakL is centerless, then \frakL [\alpha ] is split-
simple.
Proof. We begin by showing that
[\frakL [\alpha ],\frakL ] =
\left[ H[\alpha ] \oplus V[\alpha ], H \oplus
\left( \bigoplus
\beta \in [\alpha ]
\frakL \beta
\right) \oplus
\left( \bigoplus
\gamma /\in [\alpha ]
\frakL \gamma
\right) \right] \subset \frakL [\alpha ].
Clearly [H[\alpha ], H \oplus (
\bigoplus
\beta \in [\alpha ]
\frakL \beta )] + [V[\alpha ], H] \subset V[\alpha ] .
Since in case [\frakL \delta ,\frakL \tau ] \not = 0 for some \delta , \tau \in \Lambda with \delta + \tau \not = 0, the connections \{ \delta , \tau \} and
\{ \delta , \tau , - \delta \} imply [\delta ] = [\delta + \tau ] = [\tau ], we get [V[\alpha ],
\bigoplus
\beta \in [\alpha ]
\frakL \beta ] \subset \frakL [\alpha ] and
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
990 A. J. CALDERÓN MARTIN\left[ V[\alpha ],
\bigoplus
\gamma /\in [\alpha ]
\frakL \gamma
\right] = 0. (2)
This fact and Jacobi identity finally give us\left[ H[\alpha ],
\bigoplus
\gamma /\in [\alpha ]
\frakL \gamma
\right] = 0. (3)
We have showed \frakL [\alpha ] is an ideal of \frakL . Since equation (3) implies [H[\gamma ], V[\alpha ]] = 0 for any
[\gamma ] \not = [\alpha ] we get H[\alpha ] \not = 0. From here, we can also assert that \frakL [\alpha ] is a strongly split Lie algebra
with the split decomposition
\frakL [\alpha ] = H[\alpha ] \oplus
\left( \bigoplus
\beta \in [\alpha ]
\frakL \beta
\right) .
Suppose now \frakL is centerless and let us show \frakL [\alpha ] is split-simple. Consider a split-ideal I of
\frakL [\alpha ] . By equation (1) we can write I = (I \cap H[\alpha ]) \oplus (
\bigoplus
\beta \in [\alpha ]
(I \cap \frakL \beta )) with I \cap H[\alpha ] \not = 0. For any
0 \not = h \in I \cap H[\alpha ] , the fact \frakL is centerless gives us there exists \beta \in [\alpha ] such that [h,\frakL \beta ] \not = 0. From
here we get [I \cap H[\alpha ],\frakL \beta ] = \frakL \beta and so 0 \not = \frakL \beta \subset I .
Given now any \delta \in [\alpha ]\setminus \{ \pm \beta \} , the fact that \beta and \delta are connected allows us to take a connection
\{ \alpha 1, \alpha 2, . . . , \alpha n\} from \beta to \delta . Since \alpha 1, \alpha 2, \alpha 1 + \alpha 2 \in \Lambda we have [\frakL \alpha 1 ,\frakL \alpha 2 ] = \frakL \alpha 1+\alpha 2 \subset I as
consequence of \frakL \alpha 1 = \frakL \beta \subset I . In a similar way [\frakL \alpha 1+\alpha 2 ,\frakL \alpha 3 ] = \frakL \alpha 1+\alpha 2+\alpha 3 \subset I and we finally
get by following this process that \frakL \alpha 1+\alpha 2+\alpha 3+...+\alpha n = \frakL \epsilon \delta \delta \subset I for some \epsilon \delta \in \pm 1. From here we
have H[\alpha ] \subset I and as consequence, taking also into account that equation (3) allows us to assert
[H[\alpha ],\frakL \delta ] = \frakL \delta for any \delta \in [\alpha ], that V[\alpha ] \subset I . We have showed I = \frakL [\alpha ] and so \frakL [\alpha ] is split-simple.
Proposition 2.1 is proved.
Theorem 2.1. Any centerless strongly split Lie algebra is the direct sum of split-ideals, each
one being a split-simple strongly split Lie algebra.
Proof. Since we can write the disjoint union \Lambda =
\bigcup
[\alpha ]\in \Lambda /\sim
[\alpha ] we have \frakL =
\sum
[\alpha ]\in \Lambda /\sim
\frakL [\alpha ] . Let us
now verify the direct character of the sum: given x \in \frakL [\alpha ] \cap
\sum
[\beta ]\in \Lambda /\sim
\beta \nsim \alpha
\frakL [\beta ] since by equations (2) and
(3) we have [\frakL [\alpha ],\frakL [\beta ]] = 0 for [\alpha ] \not = [\beta ], we obtain
\bigl[
x,\frakL [\alpha ]
\bigr]
+
\left[ x, \sum
[\beta ]\in \Lambda /\sim
\beta \nsim \alpha
\frakL [\beta ]
\right] = 0.
From here [x,\frakL ] = 0 and so x = 0, as desired. Consequently we can write
\frakL =
\bigoplus
[\alpha ]\in \Lambda \setminus \sim
\frakL [\alpha ].
Finally, Proposition 2.1 completes the proof.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
A NOTE ON STRONGLY SPLIT LIE ALGEBRAS 991
References
1. Calderón A. J. On split Lie algebras with symmetric root systems // Proc. Indian Acad. Sci. Math. Sci. – 2008. –
118. – P. 351 – 356.
2. Kochetov M. Gradings on finite-dimensional simple Lie algebras // Acta Appl. Math. – 2009. – 108, № 1. – P. 101 – 127.
3. Nastasescu C., Van Oystaeyen F. Methods of graded rings // Lect. Notes Math. – Berlin: Springer-Verlag, 2004. –
1836.
4. Neeb K.-H. Integrable roots in split graded Lie algebras // J. Algebra. – 2000. – 225, № 2. – P. 534 – 580.
5. Schue J. R. Hilbert space methods in the theory of Lie algebras // Trans. Amer. Math. Soc. – 1960. – 95. – P. 69 – 80.
6. Stumme N. The structure of locally finite split Lie algebras // J. Algebra. – 1999. – 220. – P. 664 – 693.
Received 15.01.14,
after revision — 03.05.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| spelling | umjimathkievua-article-16112019-12-05T09:20:38Z A note on strongly split Lie algebras Зауваження щодо сильно розщеплених алгебр Лi Calderón, Martín A. J. Кальдерон, Мартін А. Дж. Split Lie algebras are maybe the most known examples of graded Lie algebras. Since an important category in the class of graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split Lie algebras $L$ and show that if $L$ is centerless, then $L$ is the direct sum of split ideals each of which is a split-simple strongly split Lie algebra. Розщепленi алгебри Лi є мабуть найбiльш вiдомим прикладом градуйованих алгебр Лi. Оскiльки важливою категорiєю в класi градуйованих алгебр є категорiя сильно градуйованих алгебр, ми вводимо (природним чином) категорiю сильно розщеплених алгебр Лi $L$ i доводимо, що у випадку, коли $L$ не має центра, $L$ є прямою сумою розщеплених iдеалiв, кожний з яких є просто-розщепленою сильно розщепленою алгеброю Лi. Institute of Mathematics, NAS of Ukraine 2018-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1611 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 7 (2018); 988-991 Український математичний журнал; Том 70 № 7 (2018); 988-991 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1611/593 Copyright (c) 2018 Calderón Martín A. J. |
| spellingShingle | Calderón, Martín A. J. Кальдерон, Мартін А. Дж. A note on strongly split Lie algebras |
| title | A note on strongly split Lie algebras |
| title_alt | Зауваження щодо сильно розщеплених алгебр Лi |
| title_full | A note on strongly split Lie algebras |
| title_fullStr | A note on strongly split Lie algebras |
| title_full_unstemmed | A note on strongly split Lie algebras |
| title_short | A note on strongly split Lie algebras |
| title_sort | note on strongly split lie algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1611 |
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