Algebraic Inverses on Lie Algebra Comultiplications

Abstract

In this note, we investigate algebraic loop structures and inverses of elements of a set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. As a symmetry phenomenon, we show that if $l{\left(1\right)}_c$ and $r{\left(1\right)}_c$ are the left and right inverses of the identity $1:\mathcal{L}\to \mathcal{L}$ on a free graded Lie algebra $\mathcal{L}$, respectively, based on the Lie algebra comultiplication ${\psi }_c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$, then we have $l\left(1\right)=l{\left(1\right)}_c$ and $r\left(1\right)=r{\left(1\right)}_c$, where $c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ is a commutator.

1. Introduction

Symmetry in mathematics is used to indicate a mathematical terminology that is invariant under transformations such as reflection, rotation and translation, and is also used to define an equivalence relation among mathematical objects; e.g., a set, a topological space or an algebraic object. Symmetry in natural sciences is generalized to mean invariance under any kind of transformation, and plays a pivotal role in quantum chemistry and theoretical physics [1,2,3] which are related to (Lie) group theory in mathematics.

The theory of Lie algebra intervenes in other areas of sciences, including different branches of physics, mathematical physics and chemistry. It is one of the active topics of current research areas and is significantly related to algebra, algebraic topology and algebraic geometry.

From the homotopical point of view, co-H-spaces, which are sometimes also called spaces with a topological comultiplication, are important objects of research; see [4,5,6,7,8,9,10,11]. According to Eckmann and Hilton, the co-H-spaces are the dual notions of H-spaces, and have been playing a pivotal role in algebraic topology for many years; see also [12,13,14,15,16].

In the present work, as an algebraic counterpart of topological comultiplications of a space [17], we consider Lie algebra comultiplications of algebraic objects and are especially interested in the algebraic loops on the set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication as a new formulation on symmetry. The main purpose of our research is to investigate the left and right inverses of elements on the algebraic structures derived from a Lie algebra comultiplication as a new formulation of symmetry. More precisely, we show that if $l{\left(1\right)}_c$ and $r{\left(1\right)}_c$ are the left and right inverses of the identity $1:\mathcal{L}\to \mathcal{L}$ on a free graded Lie algebra $\mathcal{L}$, respectively, based on the Lie algebra comultiplication ${\psi }_c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$, then we have $l\left(1\right)=l{\left(1\right)}_c$ and $r\left(1\right)=r{\left(1\right)}_c$, where $c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ is a commutator as a symmetry phenomenon based on a Lie algebra approach in mathematics, as well as the continuous symmetry and internal symmetry of particles in physics.

2. Preliminaries

We will make use of the following notation.

• $A:=\langle a_1,a_2,\dots ,a_n\rangle$ and $B:=\langle b_1,b_2,\dots ,b_m\rangle$ are graded vector spaces over a field $\mathbb{F}$ of characteristic 0.
• We denote the degree of generators $a_s,s=1,2,\dots ,n$ by $|a_s|=d_s$ with $d_1\le d_2\le \dots \le d_n$, and similarly, for $b_t,t=1,2,\dots ,m$.
• $L\left(A\right)$ and $L\left(B\right)$ are the free graded Lie algebras with the Lie brackets $[,]$ generated by A and B, respectively.

We define the coproduct $L\left(A\right)\bigsqcup L\left(B\right)$ of Lie algebras $L\left(A\right)$ and $L\left(B\right)$ as

$$L\left(A\right)\bigsqcup L\left(B\right)=L(A\oplus B).$$

As a special case, if $A=B$, then the elements of $L\left(A\right)\bigsqcup L\left(A\right)$ in either summand can be distinguished from each other using a prime from elements on the second factor. Therefore, if

$$L\left(A\right)=L(a_1,a_2,\dots ,a_n),$$

then

$$L\left(A\right)\bigsqcup L\left(A\right)=L(a_1,a_2,\dots ,a_n,a_1^{\prime },a_2^{\prime },\dots ,a_n^{\prime }).$$

Example 1.

Let $X^{\left(n\right)}$ be the nth Postnikov section of X. Then, the graded homotopy groups with rational coefficients of the Postnikov sections on the infinite complex projective space $\mathbb{C}{P}^{\infty }$ and the quaternionic projective spaces $\mathbb{H}{P}^{\infty }$ are the free graded Lie algebras with the Samelson products over the field $\mathbb{Q}$ of rational numbers; that is,

$${\pi }_{♯}{(\Omega \Sigma \mathbb{C}{P}^{\infty },\ast )}^{\left(2n\right)}\otimes \mathbb{Q}=L(a_1,a_2,\dots ,a_n)$$

and

$${\pi }_{♯}{(\Omega \Sigma \mathbb{H}{P}^{\infty },\ast )}^{\left(4m\right)}\otimes \mathbb{Q}=L(b_1,b_2,\dots ,b_m),$$

where

• Ω is the loop functor;
• Σ is the suspension functor;
• deg$\left(a_i\right)=2i$ for $i=1,2,\dots ,n$;
• deg$\left(b_j\right)=4j$ for $j=1,2,\dots ,m$.

Definition 1.

A homomorphism of Lie algebras $\psi :L\left(A\right)\to L\left(A\right)\bigsqcup L\left(A\right)$ is called a Lie algebra comultiplication (or comultiplication for short) of $L\left(A\right)$ if

$${\pi }_1\circ \psi =1_{L\left(A\right)}={\pi }_2\circ \psi ,$$

where ${\pi }_1,{\pi }_2:L\left(A\right)\bigsqcup L\left(A\right)\to L\left(A\right)$ are the projections onto the first and second factors, respectively.

For notational convenience, we will make use of polynomials to indicate the Lie brackets $[,]$; that is, $[a,b]=ab$ in the Lie algebras. We now consider the following.

Example 2.

The map $\psi :L\left(A\right)\to L\left(A\right)\bigsqcup L\left(A\right)$ of the free graded Lie algebras defined by

$$\psi \left(a_s\right)=a_s+a_s^{\prime }+P_s,$$

is a Lie algebra comultiplication, where

• $P_1=0$;
• $P_s={\sum }_{i,j}(a_i{a}_j^{\prime }+a_i^{\prime }{a}_j)$ for $s=2,3,\dots ,n$ and $i,j\in \{1,2,\dots ,n-1\}$ with $d_i+d_j=d_s$.

Indeed, it can be seen that $\psi :L\left(A\right)\to L\left(A\right)\bigsqcup L\left(A\right)$ is a Lie algebra comultiplication because

$$\begin{array}{cc}{\pi }_1\circ \psi \left(a_s\right)\hfill & ={\pi }_1(a_s+a_s^{\prime }+P_s)\hfill \\ \hfill & ={\pi }_1\left(a_s\right)+{\pi }_1\left(a_s^{\prime }\right)+{\pi }_1\left(P_s\right)\hfill \\ \hfill & =a_s+0+0\hfill \\ \hfill & =a_s\hfill \\ \hfill & =1_{L\left(A\right)}\left(a_s\right)\hfill \end{array}$$

for each $s=1,2,\dots ,n$, and similarly

$${\pi }_2\circ \psi =1_{L\left(A\right)},$$

where ${\pi }_1,{\pi }_2:L\left(A\right)\bigsqcup L\left(A\right)\to L\left(A\right)$ are the first and second projections, respectively.

Definition 2.

The element $P_s\in L\left(A\right)\bigsqcup L\left(A\right)$ is called the sth perturbation of

$$\psi :L\left(A\right)\to L\left(A\right)\bigsqcup L\left(A\right).$$

We call the sequence $P=(P_1,P_2,\dots ,P_n)$ a perturbation of the Lie algebra comultiplication ψ.

We note that if $\psi :L\left(A\right)\to L\left(A\right)\bigsqcup L\left(A\right)$ is a Lie algebra comultiplication with perturbation $P=(P_1,P_2,\dots ,P_n)$, then we have

$$\begin{array}{c}\hfill \psi \left(a_s\right)=a_s+a_s^{\prime }+P_s\end{array}$$

(1)

for $s=2,3,\dots ,n$, where $P_s$ is the sth perturbation of $\psi$ consisting of a polynomials as Lie brackets in the set $\{a_1,a_2,\dots ,a_n,a_1^{\prime },a_2^{\prime },\dots ,a_n^{\prime }\}$, each term of which contains at least one generator in $\{a_1,a_2,\dots ,a_n\}$ and at least one generator in $\{a_1^{\prime },a_2^{\prime },\dots ,a_n^{\prime }\}$. We also note that the first perturbation $P_1$ should be zero because of the degree reason of generators.

3. Algebraic Loops and Inverses

From now onward, the free graded Lie algebra $L\left(A\right)$ will be denoted as $\mathcal{L}$ for notational convenience, where $A=\langle a_1,a_2,\dots ,a_n\rangle$ is the graded vector space over $\mathbb{F}$, the field of characteristic zero. If $P=(P_1,P_2,\dots ,P_n)$ is a perturbation of a Lie algebra comultiplication

$$\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L},$$

then we sometimes denote the Lie algebra comultiplication with perturbation $P=(P_1,P_2,\dots ,P_n)$ by

$$\psi ={\psi }_P:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$$

to emphasize the perturbation P.

Let $\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ be a comultiplication of Lie algebras. Then, we define a binary operation “${+}_{\psi }$” on the set of homomorphisms of Lie algebras as follows:

Definition 3.

Let $\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ be a comultiplication. For the homomorphisms $f,g:\mathcal{L}\to \mathcal{M}$ of Lie algebras, we define a binary operation

$$f{+}_{\psi }g:\mathcal{L}\to \mathcal{M}$$

by the composite of algebra homomorphisms

$$\mathcal{L}\stackrel{\psi }{\to }\mathcal{L}\bigsqcup \mathcal{L}\stackrel{f\bigsqcup g}{\to }\mathcal{M}\bigsqcup \mathcal{M}\stackrel{\nabla }{\to }\mathcal{M};$$

that is,

$$f{+}_{\psi }g=\nabla \circ (f\bigsqcup g)\circ \psi ,$$

where ∇ is the homomorphism defined by

$$\nabla \circ (f\bigsqcup g)\left(a_s\right)=f\left(a_s\right)$$

and

$$\nabla \circ (f\bigsqcup g)\left(a_s^{\prime }\right)=g\left(a_s\right)$$

for each $a_s$ and $a_s^{\prime }$ in $\mathcal{L}$, where $s=1,2,\dots ,n$.

More specifically, if $\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ is a comultiplication with the sth perturbation defined as

$$P_s=a_i{a}_j^{\prime },$$

where $|a_i|+|a_j^{\prime }|=|a_s|$ from Equation (1), that is,

$$\psi \left(a_s\right)=a_s+a_s^{\prime }+a_i{a}_j^{\prime },$$

then

$$\left(f{+}_{\psi }g\right)\left(a_s\right)=f\left(a_s\right)+g\left(a_s\right)+f\left(a_i\right)g\left(a_j\right)$$

for $s=2,3,\dots ,n$.

Remark 1.

If

$$i_1,i_2:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$$

are the first and second inclusions, respectively, then we have

$$\psi =i_1{+}_{\psi }{i}_2:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}.$$

Let A be a set with addition “+” and trivial element $0\in A$; that is,

$$a+0=a=0+a$$

for every element $a\in A$. Then, A is said to be an algebraic loop if the following two equations

$$y+a=b$$

and

$$a+x=b$$

have the solutions $x,y$ uniquely in A for every $a,b\in A$.

Proposition 1.

Let $\psi ={\psi }_P:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ be a Lie algebra comultiplication with perturbation $P=(P_1,P_2,\dots ,P_n)$ and $\mathcal{M}$ be any Lie algebra. Then, the set $\mathrm{Hom}{(\mathcal{L},\mathcal{M})}_{\psi }$ of all Lie algebra homomorphisms $h:\mathcal{L}\to \mathcal{M}$ under the binary operation “${+}_{\psi }$” is an algebraic loop.

Proof. 

For any Lie algebra homomorphisms $f,g:\mathcal{L}\to \mathcal{M}$ in $\mathrm{Hom}{(\mathcal{L},\mathcal{M})}_{\psi }$, we must find the unique solutions $x,y\in \mathrm{Hom}{(\mathcal{L},\mathcal{M})}_{\psi }$ for the equations

$$\begin{array}{c}\hfill f{+}_{\psi }x=g\end{array}$$

(2)

and

$$\begin{array}{c}\hfill y{+}_{\psi }f=g.\end{array}$$

(3)

The Lie algebra homomorphism $x:\mathcal{L}\to \mathcal{M}$ is characterized by a set of generators of $\mathcal{L}$ by induction on the degree $|a_s|=d_s$ of $a_s$ at the perturbation level $P=(P_1,P_2,\dots ,P_n)$ for each $s=1,2,\dots ,n$. The first Equation (2) shows that

$$\begin{array}{cc}g\left(a_s\right)\hfill & =\left(f{+}_{\psi }x\right)\left(a_s\right)\hfill \\ \hfill & =\nabla \circ (f\bigsqcup x)\circ \psi \left(a_s\right)\hfill \\ \hfill & =\nabla \circ (f\bigsqcup x)(a_s+a_s^{\prime }+P_s)\hfill \\ \hfill & =f\left(a_s\right)+x\left(a_s\right)+\nabla \circ (f\bigsqcup x)\left(P_s\right);\hfill \end{array}$$

that is,

$$x\left(a_s\right)=g\left(a_s\right)-f\left(a_s\right)-\nabla \circ (f\bigsqcup x)\left(P_s\right).$$

Here $P_s$ is the sth perturbation of the Lie algebra comultiplication

$$\psi ={\psi }_P:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$$

for each $s=1,2,\dots ,n$, where $P=(P_1,P_2,\dots ,P_n)$. The induction on “s” gives the existence and uniqueness of x in Equation (2), and similarly, for the existence and uniqueness of y in Equation (3), as required. □

Let $l\left(a\right)$ and $r\left(a\right)$ be the left and right inverses of a in an algebraic loop A, respectively, under the addition “+”; i.e.,

$$l\left(a\right)+a=0=a+r\left(a\right).$$

Let

$$\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$$

be a Lie algebra comultiplication with a left inverse

$$l\left(1\right):\mathcal{L}\to \mathcal{L}$$

and a right inverse

$$r\left(1\right):\mathcal{L}\to \mathcal{L}$$

of the identity $1:\mathcal{L}\to \mathcal{L}$; that is,

$$l\left(1\right){+}_{\psi }1=0=1{+}_{\psi }r\left(1\right).$$

Then, it can be seen that

$$1=r\left(l\right(1\left)\right)$$

and

$$1=l\left(r\right(1\left)\right).$$

We now have the following:

Proposition 2.

Let $\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ be a Lie algebra comultiplication with a left inverse $l\left(1\right)$ and a right inverse $r\left(1\right)$ of $1:\mathcal{L}\to \mathcal{L}$, and let $\mathcal{M}$ be any Lie algebra. If $\alpha \in {[\mathcal{L},\mathcal{M}]}_{\psi }$, then

$$l\left(\alpha \right)=\alpha l\left(1\right)$$

and

$$r\left(\alpha \right)=\alpha r\left(1\right).$$

Proof. 

It can be shown that the following diagrams

and

are strictly commutative, where

$$\nabla :\mathcal{L}\bigsqcup \mathcal{L}\to \mathcal{L}$$

is the folding homomorphism. Therefore, we have

$$l\left(\alpha \right)=\alpha l\left(1\right)$$

and

$$r\left(\alpha \right)=\alpha r\left(1\right),$$

as required. □

From Proposition 2 above, we have

$$1=l\left(r\right(1\left)\right)=r\left(1\right)l\left(1\right)$$

and

$$1=r\left(l\right(1\left)\right)=l\left(1\right)r\left(1\right);$$

that is, $l\left(1\right)$ and $r\left(1\right)$ are Lie algebra isomorphisms.

In group theory, a commutator in a group G (that is not necessarily abelian) is an element of the form $xy{x}^{-1}{y}^{-1}$, where $x,y\in G$, and $x^{-1}$ and $y^{-1}$ are the unique inverses of x and y, respectively. As an application of the commutator in group theory, we now construct a new comultiplication based on a comultiplication

$$\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$$

by using commutators as follows.

Definition 4.

Let $\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ be a Lie algebra comultiplication on a Lie algebra $\mathcal{L}$ having a left inverse $l\left(1\right)$ and a right inverse $r\left(1\right)$ of the identity $1:\mathcal{L}\to \mathcal{L}$. Then, we can define a commutator

$$c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$$

by

$$c=\left(i_1{+}_{\psi }{i}_2\right){+}_{\psi }\left(i_{1}r\left(1\right){+}_{\psi }{i}_{2}r\left(1\right)\right),$$

where $i_1,i_2:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ are the first and second inclusion maps, respectively.

We refer to the works [18,19] for commutator theory which has been well developed in natural sciences.

Definition 5.

For a Lie algebra comultiplication $\psi :\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ and a commutator $c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$, we define a new Lie algebra comultiplication ${\psi }_c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ by

$${\psi }_c=\psi {+}_{\psi }c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}.$$

It is not difficult to show that ${\psi }_c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ is indeed a comultiplication.

Lemma 1.

The following equality

$$\nabla (l\left(1\right)\bigsqcup 1)(i_{1}r\left(1\right)+i_{2}r\left(1\right))=l\left(1\right)r\left(1\right)+r\left(1\right)$$

holds.

Proof. 

It can be shown that the following diagram

is strictly commutative, which concludes the proof. □

We now describe the main result of this paper as a new symmetry phenomenon in mathematics.

Theorem 1.

Let $c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$ be a commutator, and let $l{\left(1\right)}_c$ and $r{\left(1\right)}_c$ be the left and right inverses of the identity $1:\mathcal{L}\to \mathcal{L}$ on $\mathcal{L}$, respectively, based on the Lie algebra comultiplication ${\psi }_c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}$. Then, we have

$$l\left(1\right)=l{\left(1\right)}_c$$

and

$$r\left(1\right)=r{\left(1\right)}_c.$$

Proof. 

Let ${+}_{{\psi }_c}$ be the binary operation in ${[\mathcal{L},\mathcal{L}]}_{{\psi }_c}$ induced by the Lie algebra comultiplication

$${\psi }_c:\mathcal{L}\to \mathcal{L}\bigsqcup \mathcal{L}.$$

We note that

$$\nabla (l\left(1\right)\bigsqcup 1)\psi =l\left(1\right){+}_{\psi }1=0,$$

and the following diagram

is strictly commutative. Therefore, we have

$$\begin{array}{cc}l\left(1\right){+}_{{\psi }_c}1\hfill & =\nabla (l\left(1\right)\bigsqcup 1)\left(\psi {+}_{\psi }c\right)\hfill \\ \hfill & =\nabla (l\left(1\right)\bigsqcup 1)\psi {+}_{\psi }\nabla (l\left(1\right)\bigsqcup 1)c\hfill \\ \hfill & =\nabla \left(l\right(1)\bigsqcup 1)c.\hfill \end{array}$$

We now consider the following series of maps and a triangle:

We note that the above triangle is strictly commutative. From Lemma 1, we have

$$\begin{array}{cc}\nabla (l\left(1\right)\bigsqcup 1){\psi }_c\hfill & =\nabla (l\left(1\right)\bigsqcup 1)\left(\psi {+}_{\psi }c\right)\hfill \\ \hfill & =\nabla (l\left(1\right)\bigsqcup 1)\left(\left(i_1{+}_{\psi }{i}_2\right){+}_{\psi }\left(i_1{+}_{\psi }{i}_2\right){+}_{\psi }\left(i_{1}r\left(1\right){+}_{\psi }{i}_{2}r\left(1\right)\right)\right)\hfill \\ \hfill & =\nabla (l\left(1\right)\bigsqcup 1)\left(\left(i_1{+}_{\psi }{i}_2\right){+}_{\psi }\left(i_1{+}_{\psi }{i}_2\right)\right){+}_{\psi }\nabla (l\left(1\right)\bigsqcup 1)\left(i_{1}r\left(1\right){+}_{\psi }{i}_{2}r\left(1\right)\right)\hfill \\ \hfill & =\left(l\left(1\right){+}_{\psi }1\right){+}_{\psi }\left(l\left(1\right){+}_{\psi }1\right){+}_{\psi }\left(l\left(1\right)r\left(1\right){+}_{\psi }r\left(1\right)\right)\hfill \\ \hfill & =0{+}_{\psi }0{+}_{\psi }\left(l\left(1\right)r\left(1\right){+}_{\psi }r\left(1\right)\right)\hfill \\ \hfill & =l\left(1\right)r\left(1\right){+}_{\psi }r\left(1\right)\hfill \\ \hfill & =1{+}_{\psi }r\left(1\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

Thus,

$$l\left(1\right){+}_{{\psi }_c}1=0,$$

and so,

$$l\left(1\right)=l{\left(1\right)}_c:$$

Similarly, it can be shown that

$$1{+}_{{\psi }_c}r\left(1\right)=0$$

so that

$$r\left(1\right)=r{\left(1\right)}_c,$$

as required. □

4. Conclusions and Further Prospects

As an algebraic counterpart of topological comultiplications of a space, we have considered Lie algebra comultiplications of algebraic objects, and then, specifically explored the algebraic loop structures on the set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. We have also investigated the left and right inverses of elements on the algebraic structures derived from a Lie algebra comultiplication as a new symmetry phenomenon, which can be used in the topics of Lie algebra theory in mathematics, and the continuous symmetry and internal symmetry of particles in physics.

We do hope that our methods will be used and applied to many topics in mathematics, mathematical physics, theoretical physics and chemistry.