Research | 𝔓𝔞𝔳𝔢𝔩 𝔇𝔯𝔬𝔷𝔡𝔬𝔳

My research interests lie in the fields of Mathematics and Mathematical Physics. I mainly study the structures, inspired by physics, on the mathematical level. It has always been fascinating for me not only how effective the mathematical approach can be on the way toward cognition of Nature, but also how æsthetically logical and self-consistent it is per se. In my research work, I use mostly algebraic methods.

At the moment, I’m primarily interested in quantum groups and their relation to integrable systems.

Let us recall that an algebra $\mathscr{A}$ over a field $\mathbb{k}$ is a vector space, equipped with a bilinear map $\mu$ , that we call product or multiplication:

(1) $\begin{align*}\notag \mu: \mathscr{A} \otimes \mathscr{A} & \to \mathscr{A} & \\ (a, b) & \mapsto \mu(a, b) \eqqcolon ab \end{align*}$

Usually, we want an algebra to be associative and unitial. This means that the map $\mu$ should satisfy the following conditions:

Associativity: $(ab)c=a(bc)$ for any $a,b,c \in \mathscr{A}$ ,
Multiplicative identity: there exists an element $\mathbb{1} \in \mathscr{A}$ such that $\mathbb{1} a = a = a \mathbb{1}$ for any $a \in \mathscr{A}$ .

These two conditions can be visually represented as two commutative diagrams:

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1. Associativity of $\mathscr{A}$ ;

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2. Existence of multiplicative identity;

where we introduced the unit map

(2) $\begin{align*} \eta: \mathbb{k} & \to \mathscr{A} & \\ 1 & \mapsto \mathbb{1}. \end{align*}$

Now we can elegantly define a (coassociative counitial) coalgebra $\mathscr{C}$ over a field $\mathbb{k}$ (which is a dual of an algebra) just by “reversing arrows” on a categorial level:

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1. Coassociativity of $\mathscr{C}$ ;

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2. Existence of counit;

Here the counit is the map $\epsilon: \, \mathscr{C} \to \mathbb{k}$ , dual to the unit $\eta$ (2). From the “physical” point of view, the coproduct $\Delta: \, \mathscr{C} \to \mathscr{C} \otimes \mathscr{C}$ can be seen as an algebraic tool, allowing us to “decompose” the given system into two.

By adding one extra map $S$ (called an antipode) for compatibility purposes, we can also define a Hopf algebra $\mathscr{H}$ , which is a vector space with both algebraic and coalgebraic structures (+ antipode):

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Hopf algebra

The non-commutative and non-cocommutative Hopf algebra (a special type of it, to be more precise) is called quantum group [CP94] (the standard joke here is that the quantum group is not actually quantum and, moreover, it is not a group…)

Now, consider some Lie algebra $\mathfrak{g}$ , generated by the set $J_1, \dots, J_n$ . Its Universal Enveloping Algebra $U(\mathfrak{g})$ can be endowed with a Hopf Algebra structure, using the so-called primitive coproduct [Tji92]:

(3) $\begin{align*} \Delta_P (J_i) &= J_i \otimes \mathbb{1} + \mathbb{1} \otimes J_i, \\ \Delta_P (\mathbb{1}) &= \mathbb{1} \otimes \mathbb{1}. \end{align*}$

The primitive coproduct can be deformed in order to build the so-called quantum deformations of a Lie algebra.

So, let $(\mathscr{C},\Delta)$ be a Poisson coalgebra generated by the set $\ J_{1},\dots,J_{N}$ . Denote by $C_{i} = C_{i}\left( J_{1},\dots,J_{N} \right)$ , for $i=1,\dots,r$ , the $r$ functionally
independent Casimir functions of $\mathscr{C}$ , i.e. the elements of $\mathscr{C}$ commuting with all elements of $\mathscr{C}$ . Then, for $m>2$ by recursion one can define the left and right $m$ th coproduct $\Delta^{(m)}_{R}\colon \mathscr{C} \rightarrow \mathscr{C}^{\otimes m}$ , and $\Delta^{(m)}_{L}\colon \mathscr{C} \rightarrow \mathscr{C}^{\otimes m}$ :

(4) $\begin{align*} \Delta_{L}^{(m)} & \coloneqq \bigl({\id\otimes \dots \otimes \id}^{m-2)}\otimes \Delta\bigr)\circ\Delta^{(m-1)}, \\ \Delta^{(m)}_{R} & \coloneqq \bigl(\Delta\otimes{\id\otimes \dots \otimes \id}^{m-2)}\bigr)\circ\Delta^{(m-1)}. \end{align*}$

Then one has the following

Theorem [BCR96, BR98].
Assume we are given a Poisson coalgebra $(\mathscr{C},\Delta)$ generated by the set $\left\{ J_{1},\dots,J_{M} \right\}$ , a function $H\in\mathcal{C}^{\infty}(\mathscr{C})$ and fix $N\in\mathbb{N}$ . Define:

$H^{(N)}=\Delta^{(N)}(H)(J_1,\dots,J_M):= H(\Delta^{(N)}_{L}(J_1),\dots,\Delta^{(N)}_{L}(J_M)) \in \mathcal{C}^{\infty}(\mathscr{C}^{\otimes N}), \label{eq:HNdef}$

and left and right Casimir functions:

(5) $\begin{align*} \mathcal{F}_{m,j}:= \Delta^{(m)}_{L}(C_j)(J_1,\dots,J_M), \\ \mathcal{G}_{m,j}:= \Delta^{(m)}_{R}(C_j)(J_1,\dots,J_M), \end{align*}$

where $m=1,\dots,N, \ j=1,\dots,r$ .

Then the sets $\mathcal{L}_{\mathsf{r},N} = \{ H^{(N)}, \, \mathcal{F}_{m,j} \}_{m=1,\dots,N \atop j=1,\dots,r}$ and $\mathcal{R}_{r,N} = \{H^{(N)}, \, \mathcal{G}_{m,j} \}_{m=1,\dots,N \atop j=1,\dots,r}$ are made of Poisson-commuting functions.

The abstract theorem above states that given a Poisson coalgebra and a smooth function, one can promote it to a function on its $N$ th tensor product. This function naturally commutes with two sets of functions, the left and right Casimir functions. The function $H^{(N)}$ is then a candidate for Hamiltonian, and the left and right Casimir functions are candidate commuting invariants.

To construct a classical Hamiltonian system, one needs an $N$ -degrees-of-freedom symplectic realization of the Poisson algebra $\mathscr{C}$ , i.e. a map $\mathrm{D}_c \colon \mathscr{C} \to \Omega$ , where $\Omega\subset\mathcal{M}_{N}$ is an open subset of a symplectic manifold, such that it preserves the commutation relations of the generators of $\mathscr{C}$ .

To build a quantum system, in turn, means to build an infinite-dimensional realization in terms of operators acting on Hilbert spaces: $\mathrm{D}_q \colon\mathscr{C} \to \mathrm{Op} (\mathcal{H})$ .

Therefore, the coalgebra symmetry approach [BCR96, BR98] provides us with an elegant systematic algorithm to construct an $N$ -dimensional Liouville integrable system from a given Lie algebra. Integrable systems in contemporary mathematical and theoretical physics are so important because they usually arise as “universal” limits of huge classes of nonlinear systems. Every new system found is then a source of many new results.

In this context, my current research aspires, in particular, to study and characterize some quantum deformations of relevant Lie algebras, to construct the (super)integrable Hamiltonian systems from those quantum deformations and to extend this construction to the discrete-time systems (for which the notion of coalgebra symmetry is slightly different, see [GL23, GLT23]).

References

[BCR96] A. Ballestreros, M. Corsetti, and O. Ragnisco. N-dimensional classical integrable systems from Hopf algebras. Czech J Phys, 46(12):1153–1163, December 1996. doi:10.1007/BF01690329.

[BR98] Angel Ballesteros and Orlando Ragnisco. A systematic construction of completely integrable Hamiltonians from coalgebras. J. Phys. A: Math. Gen., 31(16):3791–3813, April 1998. doi:10.1088/0305-4470/31/16/009.

[CP94] Vyjayanthi Chari and Andrew Pressley. A guide to quantum groups. Cambridge University Press, Cambridge; New York, N.Y, 1994.

[GL23] Giorgio Gubbiotti and Danilo Latini. The sl(2,R) coalgebra symmetry and the superintegrable discrete-time systems. Phys. Scr., 98(4):045209, April 2023. arXiv:2210.17171, doi:10.1088/1402-4896/acbbb2.

[GLT23] G Gubbiotti, D Latini, and B K Tapley. Coalgebra symmetry for discrete systems. J. Phys. A: Math. Theor., 56(20):205205, May 2023. doi:10.1088/1751-8121/acc992.

[Tji92] T. Tjin. An introduction to quantized Lie groups and algebras. Int. J. Mod. Phys. A, 07(25):6175–6213, October 1992. arXiv:hep-th/9111043, doi:10.1142/ S0217751X92002805.