It is shown that a particular q-deformation of the Virasoro algebra can be interpreted in terms of the q-local field Φ(x) and the Schwinger-like point-splitted Virasoro currents, quadratic in Φ(x). The q-deformed Virasoro algebra possesses an additional index α, which is directly related to point-splitting of the currents. The generators in the q-deformed case are found to exactly reproduce the...

We extend the definition of a quantum analogue of the CalderoChapoton map defined [17]. When Q is a quiver of finite type, we prove that the algebra A H |k|(Q) generated by all cluster characters (see Definition 1) is exactly the quantum cluster algebra E H |k|(Q).

For each natural number n, poset T , and |T |–tuple of scalars Q, we introduce the ramified partition algebra P (T) n (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T | = 1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representatio...

The notions of $${{\mathcal {N}}}$$ -ideal types $$(\in , \in )$$ and \! \vee \, {q})$$ soft $${\mathcal N}_{\in }$$ -set, {N}}}_{q}$$ {N}}}_{\in {q}}$$ -subalgebra in BCK/BCI-algebra are introduced, several properties investigated. Characterizations N}$$ discussed.

The Onsager Lie algebra O is often used to study integrable lattice models. universal enveloping of admits a q-deformation Oq called the q-Onsager algebra. Recently, an was introduced alternating central extension Oq. We introduce that roughly described by following two analogies: (i) as Oq; (ii) O. call give comprehensive description

We show that intertwining operators for the discrete Fourier transform form a cubic algebra Cq, with q being root of unity. This is intimately related to other two well-known realizations algebra: Askey–Wilson and Askey–Wilson–Heun algebra.

Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and k Q its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(k Q) := Gr(k Q)/Fdim(k Q) of graded k Q-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(k Q) ≡ ModS(Q) ≡ GrL(Q) ≡ ModL(Q◦)0 ≡ QGr(k Q (n)). Here S(Q) =...

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(bg) for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conject...

Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super) algebras. Many notions from the theory of Lie (super)algebras admit “quantum” generalizations. In particular, there is a BRST operator Q (Q2 = 0) which generates the differential in the Woronowicz theory and gives information about (co)homologie...

A mapping between the operators of the bosonic oscillator and the Lorentz rotation and boost generators is presented. The analog of this map in the qdeformed regime is then applied to q-deformed bosonic oscillators to generate a q-deformed Lorentz algebra, via an inverse of the standard chiral decomposition. A fundamental representation, and the co-algebra structure, are given, and the generato...