In 2013 the first author presented his PhD at the KU Leuven with the title Symmetry and Symmetry Breaking in the Periodic Table - Towards a Group-Theoretical Classification of the Chemical Elements. The structure of the Mendeleev table is explained with an elementary particle approach and elementary particles can be classified based on a group theoretical structure and corresponding Lie algebras that are the working tools of quantum mechanics.

In this book the exposition of this idea is built from the ground up (starting with the duel that tragically ended the life of Évariste Galois, the father of group theory, followed by some elementary geometrical symmetries, the definition of a group, etc.) as it would be in a mathematical introduction to group theory. The book is however mainly written for chemistry students, spending more time on the elementary mathematics than on the elementary physics or chemistry. The ultimate goal is to explain the structure of the Mendeleev table from first principles. For the mathematics students, the organization of all the elementary particles, may not be very clear. At best, they know that the classification of all these particles is based on symmetry, and since symmetry is group theory, the group structure should be the best explanation for the classification. But it is not only groups and symmetry. Some physics and quantum physics are also needed. The atomic structure involves also angular momentum, energy levels, conservation laws, and the Schrödinger equation which defines the state of the system as a wave function that solves an eigenvalue problem for the Hamiltonian operator. It is the interplay between all these elements and how a symmetry property or a group structure is translated into properties about the spectrum of a differential operator that has to be clarified to remove the confusion. This book gives an excellent introduction to mathematical chemistry that can be perfectly used in a course about the subject. So it is certainly recommended for chemistry students. But if math students want to learn about the connection between the mathematics and the physics defining the chemistry, then this is the book they need. And if you are not a student, but a professional mathematician, who wants to be introduced to the basics of mathematical chemistry, you will be interested as well.

Let me try to explain some of the basics of all these connections starting at an elementary level of plane rotations. Let $\mathbf{a}$ be a plane vector, choose an orthogonal basis, and let $x$ and $y$ be functions mapping the vector to its coordinates $x(\mathbf{a})=a_x$ and $y(\mathbf{a})=a_y$. If $R(\omega)$ represents an operator that rotates the vector counter-clockwise over an angle $\omega\in[0,2\pi)$, then with respect to the orthogonal basis we have \[ R(\omega)\mathbf{a}=\mathbf{a}' ~~\Leftrightarrow~~ \mathbb{R}(\omega)\left[\begin{array}{c}a_x\\a_y\end{array}\right]=\left[\begin{array}{c}a'_x\\a'_y\end{array}\right],~~~ \mathbb{R}(\omega)=\left[\begin{array}{cc}\cos\omega & -\sin\omega\\ \sin\omega & \cos\omega\end{array}\right]. \] The set $\{\mathbb{R}(\omega)\in[0,2\pi)\}$ with multiplication forms the special orthogonal group SO(2) of plane rotations. It is obviously isomorphic to the group of the rotation operators $\{R(\omega):\in[0,2\pi)\}$ with composition.

If instead we keep the vector but rotate the basis vectors clockwise, then this affects the coordinate functions as follows \[ \hat{R}(\omega)\mathbf{x}= \mathbf{x}'~~\Leftrightarrow~~ \hat{\mathbb{R}}(\omega)\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{c}x'\\y'\end{array}\right],~~~ \hat{\mathbb{R}}(\omega)=\left[\begin{array}{cc}\cos\omega & \sin\omega\\ -\sin\omega & \cos\omega\end{array}\right] = \mathbb{R}(-\omega)=[\mathbb{R}(\omega)]^T=[\mathbb{R}(\omega)]^{-1} \] or $\hat{R}(\omega)[x~y]=[x'~y']=[x~y]\mathbb{R}(\omega)$.

A Taylor series expansion of $\mathbb{R}(\omega)$ defines its generator matrix $\mathbb{X}$ \[ \mathbb{R}(\omega)=\mathbb{I}+\sum_{k=1}^\infty \frac{1}{k!}(\mathbb{X}\omega)^k=\exp(\mathbb{X}\omega), \] \[ \mathbb{R}(0)=\mathbb{I}=\left[\begin{array}{cc} 1&0\\0&1\end{array}\right]=\left[\begin{array}{c}\partial_x\\\partial_y\end{array}\right][x~y],~~ \left.\frac{d^k}{d\omega^k} \mathbb{R}(\omega)\right|_{\omega=0}= \mathbb{X}^k,~~\mathbb{X}=~\left[\begin{array}{cc} 0&-1\\1&0\end{array}\right]. \] Thus if $\hat{X}=\left.\frac{d\hat{R}(\omega)}{d\omega}\right|_{\omega=0}$, then $\hat{X}[x~y]=[x~y]\mathbb{X} =[x~y]\mathbb{XI}=[x~y]\mathbb{X}\left[\begin{array}{c}\partial_x\\\partial_y\end{array}\right][x~y]$, so that $\hat{X}$ is the operator $\hat{X}=[x~y]\mathbb{X}\left[\begin{array}{c}\partial_x\\\partial_y\end{array}\right]=y\partial_x-x\partial_y$, where $\partial_x$ and $\partial_y$ represent partial derivatives. This links group elements to matrices operating on vectors and to differential operators operating on (vectors or tuples of) functions.

The differential operators allow to introduce the physical side. Consider for a moment the three-dimensional space. A rotation has the form $R(\omega\mathbf{n})$ with $\mathbf{n}$ a unit vector defining the axis of rotation (the Euler vector). Hence a rotation in 3D is defined not by 1 but by 3 parameters: 2 for the axis vector and 1 for the angle $\omega$ which can be restricted to the interval $[0,\pi)$ because $\mathbf{n}$ defines not only a direction, but also an orientation. The angular momentum vector is $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ where $\mathbf{r}=[x~y~z]$ is the position vector and $\mathbf{p}=[p_x,~p_y,~p_z]$ is the linear moment vector. The outer product says that the components of $\mathbf{L}$ are $[L_x,~L_y,~L_z]=[yp_x-zp_y,zp_x-xp_z,xp_y-yp_x]$. If we stay in the $(x,y)$-plane then $z=0$ and $\mathbf{L}$ reduces to its $z$-component. Translating this to operators we get a quantum mechanical equivalent: $\hat{{L}}_z=\hat{x}\hat{p}_y-\hat{y}\hat{p}_x=i\hslash\hat{X}$ with $\hslash=h/2\pi$ and $h$ the Planck constant. Thus we now are using the quantum mechanical position operators $[\hat{x},\hat{y},\hat{z}]=[x,y,z]$ and moment operators $[\hat{p}_x,\hat{p}_y,\hat{p}_z]=-i\hslash[\partial_x,\partial_y,\partial_z]$. Since we can also have rotations around the $x$ or $y$ axis, we have not one but three matrices $\mathbb{X}_k$, $k=x,y,z$. The rotations in three dimensional space form a special group SO(3). Moreover the three matrices $\mathbb{X}_k$, $k=x,y,z$ or equivalently, the three operators $\hat{X}_k$, $k=x,y,z$ generate a Lie algebra ${\frak so}(3)$. That means that it has a composition defined by Lie brackets $[\hat{X}_i,\hat{X}_j]=\hat{X}_i\hat{X}_j-\hat{X}_j\hat{X}_i=\epsilon_{ijk} \hat{X}_k$ with $\epsilon_{ijk}\in\{0,+1,-1\}$ the structure constants for the Lie algebra. ($\epsilon_{ijk}$ is $+1$ or $-1$ if $ijk$ is an even or an odd permutation of $xyz$, and it is zero when two of the three indices are equal.) This gives the link with Lie algebras.

Next we need a link between symmetries and conservation laws, and between the wave function and eigenvalue problems. A physical invariant under space translation results in a conservation of the linear momentum. An invariant under time delays translates in a conservation of energy, and rotational symmetry defines the conservation of angular momentum.

The invariant linear momentum is $\hat{p}{}^2=\hat{\mathbf{p}} \cdot\hat{\mathbf{p}}=-\hslash^2[\partial_x^2+\partial_y^2+\partial_z^2]$, which gives the wave operator which defines the kinetic part of the Hamiltonian $\hat{\mathcal{H}}=\hat{p}^2/2m$ ($m$ is the mass of the particle). If we choose for a rotation around the $z$-axis, we have the eigenvalue problem $\hat{L}_z|\Psi\rangle=\lambda|\Psi\rangle$. This $\Psi$ is the wave function defining the state, written here as a "ket" which is to be understood as a column vector (the corresponding row is called a "bra" and denoted as $\langle\Psi|$, so that the inner product is a "braket" $\|\Psi\|^2=\langle\Psi|\Psi\rangle$). The eigenvalues are $\lambda=\hslash m_l$ with $m_l$ taking all integer values (magnetic quantum numbers). These integers are reflecting the periodicity in $\omega$. This gives a model of an electron confined to a circular ring with only one degree of freedom: the rotation angle.

The invariant for angular momentum is $\hat{L}{}^2=\hat{\mathbf{L}}\cdot\hat{\mathbf{L}}$. The eigenvalues of this operator are again discrete: $l(l+1)\hslash^2$ with $l\in\mathbb{N}/2$. In 3D the corresponding eigenstate $|\Psi\rangle$ will depend on two parameters if it is confined to a spherical shell. We denote the eigenstate as $|l,m_l\rangle$ with $m_l$ as in the circular case. Thus $\hat{L}{}^2|l,m_l\rangle=l(l+1)\hslash^2|l,m_l\rangle$. However, combining both eigenvalue problems restricts $m_l$ to $\{-l,\ldots,l\}$. The operator $\hat{L}{}^2$ commutes with all generators of the Lie algebra ${\frak so}(3)$ and is therefore called a Casimir operator, the only one in the case of ${\frak so}(3)$.

The time independent Schrödinger equation (i.e. assuming no potential energy) introduces energy $E$ as an eigenvalue of the problem $\hat{\mathcal{H}}|\Psi\rangle=E|\Psi\rangle$, where $\hat{\mathcal{H}}$ is the Hamiltonian operator given by $\hat{\mathcal{H}}=\hat{L}{}^2/(2mr^2)$ if the particle with mass is restricted to the spherical shell wit radius $r$. Hence the eigenvalues are $E_l=l(l+1)\hslash^2/(2mr^2)$ where $l$ refers to the successive shells, which are traditionally indicated by $s,p,d,f,...$ for $l=0,1,2,3,...$.

If $\hat{Y}$ is an operator commuting with $\hat{\mathcal{H}}$, then its expected value $\langle\hat{Y}\rangle=\langle\Psi|\hat{Y}|\Psi\rangle$ will not vary in time: $\partial_t\langle\hat{Y}\rangle=0$. Since the Hamiltonian commutes with itself, the energy is conserved. Solving the Hamiltonian eigenvalue problem gives discrete values for the energy $E$ and the corresponding eigenspace can be one dimensional or of higher dimension, corresponding to a nondegenerate or degenerate case respectively.

The operators $\hat{X}_i$ that commute with the Hamiltonian, thus satisfy $\hat{\mathcal{H}}\hat{X}_i|\Psi\rangle=E\hat{X}_i|\Psi\rangle$ and that map the eigenvector $|\Psi\rangle$ onto a multiple of itself (as on the nondegenerate case) are called Cartan generators (denoted $\hat{H}_i$). They generate in the general case a maximal abelian subalgebra. In our case $\hat{L}_z$ is such a Cartan generator. The remaining generators can be recombined into a set of Weyl generators. In our example there are two: $\hat{L}_\pm=\hat{L}_x\pm\hat{L}_y$, which are also called ladder operators. They shift the eigenvalues. Suppose $|l,m_l\rangle$ denotes the common eigenvector $|\Psi\rangle$ of $\hat{L}{}^2$ and $\hat{L}_z$ with eigenvalues $\hslash m_l$ and $l(l+1)\hslash^2$ respectively, then for example $\hat{L}_z\hat{L}{}_\pm^k|l,m_l\rangle=(m_l\pm k)\hslash\hat{L}{}_\pm^k|l,m_l\rangle$. Thus $\hat{L}{}_\pm$ shift the eigenvalue of $\hat{L}_z$ up or down the ladder with one unit. If the Casimir operator has eigenvalues $c_\mu$: $\hat{C}_\mu|c_\mu;h_i\rangle = c_\mu|c_\mu;h_i\rangle$, and the Cartan operator has eigenvalues $h_i$: $\hat{H}_i|c_\mu;h_i\rangle=h_i|c_\mu;h_i\rangle$ then a Weyl operator $\hat{E}_\alpha$ satisfies $\hat{H}_i\hat{E}_\alpha|c_\mu;h_i\rangle=(h_i+\alpha_i)\hat{E}_\alpha|c_\mu;h_i\rangle$ and thus $\hat{E}_\alpha|c_\mu;h_i\rangle\sim|c_\mu;h_i+\alpha_i\rangle$.

All this is just a brief summary of the introductory part I of the book building up the elementary quantum mechanical models in two and three dimensions. It ends with a scholium chapter giving a taste of the $n$-dimensional case. To arrive at the structure of the Mendeleev table, much more is needed. Part II introduces the dynamics and the "zoo of elementary particles" (pions, kaons, mesons,...) which requires the introduction of charge, spin, strangeness,... The Hamiltonian now involves a part for the potential energy. The group structure is extended to the unitary matrices and operators of U(3) (with 9 generators $\mathbb{X}_i$) and the special unitary subgroup SU(3) generated by three of them. One can again define a Lie algebra ${\frak su}(3)$ from these generators with corresponding Cartan and Weyl operators and Casimir invariants. This algebra is now much richer and different subalgebras can be defined. For example the reduction SU(3)$\to$SO(3) is called a symmetry breaking. One can think of it as a projection from the complex plane onto the real axis. More involved spinor operators $\hat{S}_i$ can be defined, associated with the famous Pauli matrices \[ \sigma_x=\left[\begin{array}{cc}1&0\\0&-1\end{array}\right],~~ \sigma_y=\left[\begin{array}{cc}0&-i\\i&0\end{array}\right],~~ \sigma_z=\left[\begin{array}{cc}0&1\\1&0\end{array}\right]. \] A spinor is like a rotation taking place on a Möbius band: after a rotation over $2\pi$ one ends up with the negative vector. This explains, or is explained by, the introduction of complex numbers: $i^2=-1$. One has to rotate over $4\pi$ to arrive at the original position. This explains doubling effects. The spinors generate SU(2) which is the double covering group for SO(3) because every rotation in 3D is the image of two elements in SU(2).

Newtonian mechanics and Kepler's rules are applied to derive the orbit of an electron around the kernel in classical mechanics. The quantum mechanical analog of the so called LRL (Laplace-Runge-Lenz) vector $\mathbf{M}$ (the vector defining the larger half axis of the elliptic trajectory) consists of three operators $\hat{M}_k$. Together with the $\hat{L}_k$ operators they generate the Lie algebra ${\frak so}(4)$.

Furthermore this part II has historical notes about the Kepler problem, the LRL vector, and the attempts to bring order in the particle zoo by e.g. Gell-Mann in 1960's. It also introduces root diagrams, which are 2D grids representing quantum numbers. Projections onto certain lines represent degeneracies, that are reductions to subalgebras. On such lines some parameter (like quantum number or spin) is constant. A root diagram catches the essentials of a Lie algebra. A root diagram for ${\frak su}(3)$ for example places the Cartan generators at the center of a regular hexagon and the six vertices describe the actions of the six Weyl generators.

Part III is about spectral generating symmetries and thus arrives at the ultimate goal of explaining the Mendeleev table. We can characterize an eigenstate by four quantum numbers: the principal quantum number $n$ referring to energy $E_n$, the orbital quantum number $l\in\{0,\ldots,n-1\}$ referring to the shell and its angular momentum, the magnetic quantum number $m_l\in\{-l,\ldots,l\}$, and $m_s\in\{\pm1/2\}$, the spin (up or down). For a true spectrum generating symmetry one has to consider transformations from eigenstate $|nl\rangle$ to eigenstate $|n'l\rangle$ with $n'\ne n$. Since $n$ refers to the energy levels, the energy changes, the Hamiltonian will not be invariant under these transformations and one has to consider the radial wave equation. As a consequence the previous group SO(3) has to be replaced by the pseudo orthogonal group SO(2,1). Note that SO(2,1) is not compact and the Lie algebra ${\frak so}(2,1)$ has infinite dimensional unitary representations, corresponding to orbital numbers $n=l+1,l+2,\ldots$. While the SO(3) transformations leave the sphere invariant, the SO(2,1) will leave a hyperboloid invariant. The 2,1 refers to the signature $(++-)$ of the hyperboloid. SO(2,1) has three generators $\hat{Q}_k$. Choosing the third one as an analog of $\hat{L}_z$, then it has to eigenvalues $n=l+1+s$, $s=0,1,2,...$ corresponding to energy levels $E_n=-mZ^2e^4/(8h^2\epsilon_0^2 n^2)$, $n=1,2,\ldots$ with $Z$ the atomic number (which is 1 for hydrogen which has only one electron), $e$ is the electron charge, $\epsilon_0$ vacuum permittivity. To describe the eigenstates $|nlm\rangle$, root diagrams for the Lie algebra ${\frak su}(4,2)$ are needed, which are now 3D instead of 2D grids, and one has ladder operators that increase or decrease each of the $n,l$ or $m$ parameters separately.

In the penultimate chapter the structure of the Mendeleev table is finally analysed. The group structure proposed by the authors is caught in the following symmetry breakings $SO(4,2)\otimes SU(2) \supset SO(3,2)\otimes SU(2) \supset SO'(4)\otimes SU(2)$. The first is the overall symmetry group with SU(2) corresponding to the spin. All possible $(n,l)$ couples will represent all the chemical elements corresponding to a basis for the infinite-dimensional unitary representation (unirep) of the group. The reduction to SO(3,2) corresponds to the period doubling because the elements split into two sets where $n+l$ is either even or odd. The last chapter explains SO'(4). Using a metaphor of a triangular chessboard with squares $\{(n,l): n=1,2,3\ldots; l=0,1,\ldots,n-1\}$. The moves on the chessboard correspond to operators. For example the rook can move vertical (operators in SO(4)) and horizontal (operators in SO(2,1)) so that this corresponds to SO(4)$\otimes$SO(2,1). Analogously one can have operators corresponding to the king, queen, knight, bishop, and pawn pieces, each belonging to specific groups. The diagonal moves of the bishop along diagonals corresponds to so-called Madelung $n\pm l$ rules and the corresponding Madelung operators are in SO(3,2). However, since there is an upward and a downward sloping diagonal, this should correspond to a further symmetry breaking. This is not obtained by a standard reduction of SO(3,2) to SO(3,1). Instead one has to combine left-right reflections with up-down reflection operators to get the separate diagonals. This required the introduction of a new Lie algebra ${\frak so}'(4,2)$ and one is forced to give up linearity.

Clearly the previous survey just lifts the corner of the veil that covers the magical world of elementary particles and atomic structures. All the details of this story can be found in the book, explained with much care. Do realise that it scratches only the surface: orbitals of particles moving around a nucleus. It remains away from relativity theory, supersymmetry, strings, and membranes. Moreover, it is not just the dry mathematics, but it is brought with much imagination. I mentioned already some of the historical excursions, and there is the chessboard in the last chapter. With this chessboard, the authors refer to Lewis Carroll's Alice in Wonderland, as they do throughout the book, borrowing the wonderful illustrations by John Tenniel. In principle no previous knowledge is required (73 pages with appendices recall the necessary background or work out some of the longer computations), it is still hard work for someone who is not already a bit familiar with the subject or has only some background in either mathematics or in chemistry. Nevertheless, it is a most fascinating story marrying mathematics, physics and chemistry that is a joy to read about, or work in. It is abstract mathematics and yet it describes some basic elements of physical reality.