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| In [[mathematics]], a '''graded Lie algebra''' is a [[Lie algebra]] endowed with a [[graded vector space|gradation]] which is compatible with the [[Commutator|Lie bracket]]. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of [[Cartan decomposition]] endows any [[semisimple Lie algebra]] with the structure of a graded Lie algebra. Any [[parabolic Lie algebra]] is also a graded Lie algebra.
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| A '''graded Lie superalgebra'''<ref>The "super" prefix for this is not entirely standard, and some authors may opt to omit it entirely in favor of calling a graded Lie superalgebra just a ''graded Lie algebra''. This dodge is not entirely without warrant, since graded Lie superalgebras may have nothing to do with the algebras of [[supersymmetry]]. They are only super insofar as they carry a '''Z'''/2'''Z''' gradation. This gradation occurs naturally, and not because of any underlying superspaces. Thus in the sense of [[category theory]], they are properly regarded as ordinary non-super objects.</ref> extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily [[anticommutative]]. These arise in the study of [[derivation (abstract algebra)|derivation]]s on [[graded algebra]]s, in the [[deformation theory]] of [[Murray Gerstenhaber|M. Gerstenhaber]], [[Kunihiko Kodaira]], and [[D. C. Spencer]], and in the theory of [[Lie derivative]]s.
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| A '''supergraded Lie superalgebra'''<ref>In connection with [[supersymmetry]], these are often called just ''graded Lie superalgebras'', but this conflicts with the previous definition in this article.</ref> is a further generalization of this notion to the category of [[superalgebra]]s in which a graded Lie superalgebra is endowed with an additional super '''Z'''/2'''Z'''-gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supersymmetric) setting, and then tensorizes to obtain the [[supersymmetry|supersymmetric]] analog.<ref>Thus supergraded Lie superalgebras carry a ''pair'' of '''Z'''/2'''Z'''-gradations: one of which is supersymmetric, and the other is classical. [[Pierre Deligne]] calls the supersymmetric one the ''super gradation'', and the classical one the ''cohomological gradation''. These two gradations must be compatible, and there is often disagreement as to how they should be regarded. See [http://www.math.ias.edu/QFT/fall/bern-appen1.ps Deligne's discussion] of this difficulty.</ref>
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| Still greater generalizations are possible to Lie algebras over a class of [[braided monoidal category|braided monoidal categories]] equipped with a [[coproduct]] and some notion of a gradation compatible with the braiding in the category. For hints in this direction, see [[Lie algebra#Category theory definition]].
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| == Graded Lie algebras ==
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| In its most basic form, a graded Lie algebra is an ordinary Lie algebra <math>{\mathfrak g}</math>, together with a gradation of vector spaces:
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| :<math>{\mathfrak g}=\bigoplus_{i\in{\mathbb Z}} {\mathfrak g}_i</math> (1)
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| such that the Lie bracket respects this gradation:
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| :<math>[{\mathfrak g}_i,{\mathfrak g}_j]\subseteq {\mathfrak g}_{i+j}.</math> (2)
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| The [[universal enveloping algebra]] of a graded Lie algebra inherits the grading.
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| === Examples ===
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| ==== sl(2) ====
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| For example, the Lie algebra sl(2) of [[trace-free]] 2x2 [[matrix (mathematics)|matrices]] is graded by the generators:
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| :<math>
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| X=\left(\begin{matrix}0&1\\0&0\end{matrix}\right),\quad Y=\left(\begin{matrix}0&0\\1&0\end{matrix}\right),
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| </math> and
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| :<math>
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| H=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right).
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| </math>
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| These satisfy the relations [''X'',''Y''] = ''H'', [''H'',''X''] = 2''X'', [''H'',''Y''] = -2''Y''. Hence with
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| ''g''<sub>-1</sub> = span(''X''), ''g''<sub>0</sub> = span(''H''), and ''g''<sub>1</sub> = span(''Y''),
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| the decomposition sl(2) = ''g''<sub>-1</sub> + ''g''<sub>0</sub> + ''g''<sub>1</sub> presents sl(2) as a graded Lie algebra.
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| ==== Free Lie algebra ====
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| {{main|Free Lie algebra}}
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| The [[free Lie algebra]] on a set ''X'' naturally has a grading, given by the minimum number of terms needed to generate the group element. This arises for example as the associated graded Lie algebra to the [[lower central series]] of a [[free group]].
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| === Generalizations ===
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| If Γ is any [[commutative monoid]], then the notion of a '''Γ-graded Lie algebra''' generalizes that of an ordinary ('''Z'''-) graded Lie algebra so that the defining relations (1) and (2) hold with the integers '''Z''' replaced by Γ. In particular, any semisimple Lie algebra is graded by the root spaces of its [[adjoint representation of a Lie algebra|adjoint representation]].
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| == Graded Lie superalgebras ==
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| A '''graded Lie superalgebra''' over a field ''k'' (not of [[characteristic of a field|characteristic]] 2) consists of a [[graded vector space]] ''E'' over ''k'', along with a [[bilinear form|bilinear]] '''bracket''' operation
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| ::<math>[-,-] : E\otimes_k E\rightarrow E</math>
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| such that the following axioms are satisfied.
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| :*[-,-] respects the gradation of ''E'':
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| ::<math>[E_i,E_j]\subseteq E_{i+j}</math>.
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| :*(''Symmetry''.) If ''x'' ε E<sub>i</sub> and ''y'' ε E<sub>j</sub>, then
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| ::<math>[x,y]=-(-1)^{ij}\,[y,x]</math>
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| :*(''Jacobi identity''.) If ''x'' ε E<sub>i</sub>, ''y'' ε E<sub>j</sub>, and ''z'' ε E<sub>k</sub>, then
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| ::<math>(-1)^{ik}[x,[y,z]]+(-1)^{ij}[y,[z,x]]+(-1)^{jk}[z,[x,y]]=0</math>.
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| ::(If ''k'' has characteristic 3, then the Jacobi identity must be supplemented with the condition <math>[x,[x,x]] = 0</math> for all ''x'' in ''E''<sub>odd</sub>.)
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| Note, for instance, that when ''E'' carries the trivial gradation, a graded Lie superalgebra over ''k'' is just an ordinary Lie algebra. When the gradation of ''E'' is concentrated in even degrees, one recovers the definition of a ('''Z'''-) graded Lie algebra.
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| === Examples and Applications ===
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| The most basic example of a graded Lie superalgebra occurs in the study of derivations of graded algebras. If ''A'' is a [[graded algebra|graded ''k''-algebra]] with gradation
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| :<math>A=\bigoplus_{i\in{\mathbb Z}} A_i</math>,
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| then a graded ''k''-derivation ''d'' on ''A'' of degree ''l'' is defined by
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| #''dx'' = 0 for ''x'' ε ''k'',
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| #''d'' : ''A''<sub>i</sub> → ''A''<sub>i+l</sub>, and
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| #''d''(''xy'') = (''dx'')''y'' + (-1)<sup>il</sup>''x''(''dy'') for ''x'' ε ''A''<sub>i</sub>.
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| The space of all graded derivations of degree ''l'' is denoted by Der<sub>l</sub>(''A''), and the direct sum of these spaces
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| :<math>\hbox{Der}(A)=\bigoplus_l \hbox{Der}_l(A)</math>
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| carries the structure of an ''A''-module. This generalizes the notion of a derivation of commutative algebras to the graded category.
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| On Der(''A''), one can define a bracket via:
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| :[''d'',δ]=''d'' δ - (-1)<sup>ij</sup>δ ''d'', for ''d'' ε Der<sub>i</sub>(''A'') and δ ε Der<sub>j</sub>(''A'').
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| Equipped with this structure, Der(''A'') inherits the structure of a graded Lie superalgebra over ''k''.
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| '''Further examples:'''
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| * The [[Frölicher-Nijenhuis bracket]] is an example of a graded Lie algebra arising naturally in the study of [[connection (mathematics)|connections]] in [[differential geometry]].
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| * The [[Nijenhuis-Richardson bracket]] arises in connection with the deformations of Lie algebras.
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| === Generalizations ===
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| The notion of a graded Lie superalgebra can be generalized so that their grading is not just the integers. Specifically, a '''signed semiring''' consists of a pair (Γ, ε) where Γ is a [[semiring]] and ε : Γ → '''Z'''/2'''Z''' is a [[homomorphism]] of additive groups. Then a graded Lie supalgebra over a signed semiring consists of a vector space ''E'' graded with respect to the additive structure on Γ, and a bilinear bracket [-,-] which respects the grading on ''E'' and in addition satisfies:
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| # <math>[x,y] = (-1)^{\epsilon(\hbox{deg}\ x)\epsilon(\hbox{deg}\ y)}[y,x]</math> for all [[homogeneous element]]s ''x'' and ''y'', and
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| # <math>(-1)^{\epsilon(\hbox{deg}\ x)\epsilon(\hbox{deg}\ z)}[x,[y,z]] + (-1)^{\epsilon(\hbox{deg}\ y)\epsilon(\hbox{deg}\ x)}[y,[z,x]] + (-1)^{\epsilon(\hbox{deg}\ z)\epsilon(\hbox{deg}\ y)}[z,[x,y]]=0.</math>
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| '''Further examples:'''
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| *A [[Lie superalgebra]] is a graded Lie superalgebra over the signed semiring ('''Z'''/2'''Z''',ε) where ε is the identity endomorphism for the additive structure on the ring '''Z'''/2'''Z'''.
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| ==Notes==
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| <references/>
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| == References ==
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| * Nijenhuis, A., and Richardson, R. W. Jr., "Cohomology and deformations in graded Lie algebras", ''Bull. AMS'' '''72''' (1966), 1-29.
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| == See also ==
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| * [[Differential graded Lie algebra]]
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| * [[Graded (mathematics)]]
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| {{DEFAULTSORT:Graded Lie Algebra}}
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| [[Category:Lie algebras]]
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