Graded Lie algebra - Citizendia

In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a graded vector space is a type of Vector space that includes the extra structure of gradation, meaning that it can be composed into the 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. The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory In Mathematics, a Lie algebra is semisimple if it is a Direct sum of Simple Lie algebras i Any parabolic Lie algebra is also a graded Lie algebra. In the theory of Algebraic groups, a Borel subgroup of an Algebraic group G is a maximal Zariski closed and connected solvable

A graded Lie superalgebra^[1] extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. In mathematics anticommutativity refers to the property of an operation being anticommutative, i These arise in the study of derivations on graded algebras, in the deformation theory of M. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Gerstenhaber, Kunihiko Kodaira, and D. C. Spencer, and in the theory of Lie derivatives. was a Japanese Mathematician known for distinguished work in Algebraic geometry and the theory of Complex manifolds and as the founder of the Japanese Donald Clayton Spencer ( April 25 1912 - December 23 2001) was an American Mathematician, known for major work on Deformation In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the

A supergraded Lie superalgebra^[2] is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super Z/2Z-gradation. In Mathematics and Theoretical physics, a superalgebra is a Z 2- Graded algebra. These arise when one forms a graded Lie superalgebra in a classical (non-supersymmetric) setting, and then tensorizes to obtain the supersymmetric analog. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that ^[3]

Still greater generalizations are possible to Lie algebras over a class of braided monoidal categories equipped with a coproduct and some notion of a gradation compatible with the braiding in the category. In Mathematics, a braided monoidal category is a Monoidal category C equipped with a braiding; that is there is a Natural isomorphism In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological For hints in this direction, see Lie algebra#Category theory definition. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

1 Graded Lie algebras
2 Graded Lie superalgebras
- 2.1 Examples and Applications
- 2.2 Generalizations
3 Notes
4 References

Graded Lie algebras

In its most basic form, a graded Lie algebra is an ordinary Lie algebra ${\mathfrak g}$ , together with a gradation of vector spaces:

${\mathfrak g}=\bigoplus_{i\in{\mathbb Z}} {\mathfrak g}_i$ (1)

such that the Lie bracket respects this gradation:

$[{\mathfrak g}_i,{\mathfrak g}_j]\subseteq {\mathfrak g}_{i+j}.$ (2)

For example, the Lie algebra sl(2) of trace-free 2x2 matrices is graded by the generators:

$X=\left(\begin{matrix}0&1\\0&0\end{matrix}\right),\quad Y=\left(\begin{matrix}0&0\\1&0\end{matrix}\right),$ and

$H=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right)$ . TRACE ( Transition Region and Coronal Explorer) is a NASA space telescope designed to investigate the connections between fine-scale magnetic fields and In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

These satisfy the relations [X,Y] = H, [H,X] = 2X, [H,Y] = -2Y. Hence with g_-1 = span(X), g₀ = span(H), and g₁ = span(Y), the decomposition sl(2) = g_-1 + g₀ + g₁ presents sl(2) as a graded Lie algebra.

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 Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In particular, any semisimple Lie algebra is graded by the root spaces of its adjoint representation. In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its

Graded Lie superalgebras

A graded Lie superalgebra over a field k (not of characteristic 2) consists of a graded vector space E over k, along with a bilinear bracket operation

$[-,-] : E\otimes_k E\rightarrow E$

such that the following axioms are satisfied. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, a graded vector space is a type of Vector space that includes the extra structure of gradation, meaning that it can be composed into the

[-,-] respects the gradation of E:

$[E_i,E_j]\subseteq E_{i+j}$ .

(Symmetry. ) If x ε E_i and y ε E_j, then

$[x,y]=-(-1)^{ij}\,[y,x]$

(Jacobi identity. ) If x ε E_i, y ε E_j, and z ε E_k, then

( - 1) i k [x,[y, z]] + ( - 1) i j [y,[z, x]] + ( - 1) j k [z,[x, y]] = 0

(If k has characteristic 3, then the Jacobi identity must be supplemented with the condition [x,[x,x]] = 0 for all x in E_odd. )

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.

Examples and Applications

The most basic example of a graded Lie superalgebra occurs in the study of derivations of graded algebras. If A is a graded k-algebra with gradation

$A=\bigoplus_{i\in{\mathbb Z}} A_i$ ,

then a graded k-derivation d on A of degree l is defined by

dx = 0 for x ε k,
d : A_i → A_i+l, and
d(xy) = (dx)y + (-1)^ilx(dy) for x ε A_i. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure

The space of all graded derivations of degree l is denoted by Der_l(A), and the direct sum of these spaces

$\hbox{Der}(A)=\bigoplus_l \hbox{Der}_l(A)$

carries the structure of an A-module. This generalizes the notion of a derivation of commutative algebras to the graded category.

On Der(A), one can define a bracket via:

[d,δ]=d δ - (-1)^ijδ d, for d ε Der_i(A) and δ ε Der_j(A).

Equipped with this structure, Der(A) inherits the structure of a graded Lie superalgebra over k.

Further examples:

The Frölicher-Nijenhuis bracket is an example of a graded Lie algebra arising naturally in the study of connections in differential geometry. In Mathematics, the Frölicher-Nijenhuis bracket is an extension of the Lie bracket of Vector fields to Vector-valued differential forms on a In Geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry
The Nijenhuis-Richardson bracket arises in connection with the deformations of Lie algebras. In Mathematics, the algebraic bracket or Nijenhuis-Richardson bracket is a Graded Lie algebra structure on the space of alternating multilinear forms

Generalizations

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/2Z is a homomorphism of additive groups. In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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:

$[x,y] = (-1)^{\epsilon(\hbox{deg}\ x)\epsilon(\hbox{deg}\ y)}[y,x]$ for all homogeneous elements x and y, and
$(-1)^{\epsilon(\hbox{deg}\ x)\epsilon(\hbox{deg}\ z)}[x,[y,z]] + (-1)^{\epsilon(\hbox{deg}\ y)\epsilon(\hbox{deg}\ x)}[y,[z,x]] + (-1)^{\epsilon(\hbox{deg}\ z)\epsilon(\hbox{deg}\ y)}[z,[x,y]]=0.$

Further examples:

A Lie superalgebra is a graded Lie superalgebra over the signed semiring (Z/2Z,ε) where ε is the identity endomorphism for the additive structure on the ring Z/2Z. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z 2- grading.

Notes

^ 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. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that They are only super insofar as they carry a Z/2Z 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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
^ In connection with supersymmetry, these are often called just graded Lie superalgebras, but this conflicts with the previous definition in this article. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that
^ Thus supergraded Lie superalgebras carry a pair of Z/2Z-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. Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician. These two gradations must be compatible, and there is often disagreement as to how they should be regarded. See Deligne's discussion of this difficulty.

References

Nijenhuis, A. , and Richardson, R. W. Jr. , "Cohomology and deformations in graded Lie algebras", Bull. AMS 72 (1966), 1-29.

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