Lie algebra object



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Category theory



The notion of Lie algebra may be formulated internal to general linear monoidal categories (tensor categories). This general definition of Lie algebra objects internal to tensor categories subsumes variants of Lie algebras such as super Lie algebras.


As internal Lie algebras

Consider a commutative unital ring kk, and a (strict for simplicity) symmetric monoidal kk-linear category (𝒞,,1)(\mathcal{C},\otimes,1) with braiding τ\tau.

A Lie algebra object in (𝒞,,1,τ)(\mathcal{C},\otimes,1,\tau) is

  1. an object

    L𝒞 L \in \mathcal{C}
  2. morphism (the Lie bracket)

    [,]:LLL [-,-] \;\colon\; L \otimes L \to L

such that the following conditions hold:

  1. Jacobi identity:

    [,[,]]+[,[,]](id Lτ L,L)(τid L)+[,[,]](τ L,Lid L)(id Lτ L,L)=0 \left[-,\left[-,-\right]\right] + \left[-,\left[-,-\right]\right] \circ(id_L\otimes\tau_{L,L}) \circ(\tau\otimes id_L) + \left[-,\left[-,-\right]\right] \circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0
  2. skew-symmetry:

    +[,] +[,]τ L,L =+0 \begin{aligned} & \phantom{+} [-,-] \\ & + [-,-]\circ \tau_{L,L} \\ & = \phantom{+} 0 \end{aligned}

Equivalently, in string diagram-notation in the ambient tensor category, the Lie action property looks as follows:

ρ(f(x,y),z)=ρ(y,ρ(x,z))ρ(x,ρ(y,z)) f=[,]Liebracket ρ([x,y],z)=ρ(y,ρ(x,z))ρ(x,ρ(y,z))Lieactionproperty ρ=[,]adjointaction [[x,y],z]=[y,[x,z]]+[x,[y,z]]Jacobiidentity \begin{aligned} \Leftrightarrow & \;\;\;\;\; \rho(f(x,y),z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z)) \\ \underset{ {f = [-,-]} \atop {Lie\;bracket} }{ \Leftrightarrow } & \;\;\;\;\; \underset{ {Lie\;action\;property} }{ \underbrace{ \rho([x,y],z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z)) } } \\ \underset{ {\rho = -[-,-]} \atop {adjoint\;action} }{ \Leftrightarrow } & \;\;\;\;\; \underset{ {Jacobi\;identity} }{ \underbrace{ [[x,y],z] \;=\; - [y,[x,z]] + [x,[y,z]] } } \end{aligned}

where the last line shows the equivalence to the Jacobi identity on the Lie algebra object itself in the case that the Lie action is the adjoint action.

In the language of Jacobi diagrams this is called the STU-relation. and is the reason behind the existence of Lie algebra weight systems in knot theory. For more see also at metric Lie representation.

As algebras over the Lie operad

Equivalently, Lie algebra objects are the algebras over an operad over a certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.


Examples of types of Lie algebra objects:

If kk is the ring \mathbb{Z} of integers and 𝒞=\mathcal{C} = kkMod = Ab is the category of abelian groups equipped with the tensor product of abelian groups, then a Lie algebra object is called a Lie ring.

If kk is a field and 𝒞=\mathcal{C} = Vect is the category of vector spaces over kk equipped with the tensor product of vector spaces then a Lie algebra object is an ordinary_Lie k-algebra.

If kk is a field and 𝒞\mathcal{C} = sVect is the category of super vector spaces over kk, then a Lie algebra object is a super Lie algebra.

Last revised on December 7, 2019 at 07:41:03. See the history of this page for a list of all contributions to it.