Contractions of Lie algebras are combined with the classical matrix method of Gel’fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudounitary Lie algebras Iu(p, q). This procedure is extended to contractions of Iu(p, q) isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Li...

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...

Proceeding from the nonstandard q-deformed algebras U ′ q(son+1) and their finitedimensional representations in a q-analog of the Gel’fand-Tsetlin basis, we obtain, by means of the contraction procedure, the corresponding q-deformed inhomogeneous algebras U ′ q(ison) in a uniform fashion for all n ≥ 2 as well as their infinite-dimensional representations.

We define the concepts of representable and abstract sequential Q-algebra, which are generalizations of the (relational) Q-algebras in [10]. Just as in that paper, we then prove that the two concepts coincide. In the following section we recall the concept of observation space and note that all complex algebras of observation spaces are representable sequential algebras. Finally we give an unco...

APR tilts for path algebra kQ can be realized as the mutation of quiver Q in $${\mathbb Z}Q$$ with respect to translation. In this paper, we show that have similar results quadratic dual truncations n-translation algebras; is, under certain condition, n-APR such algebras are $$\tau $$ -mutations. For -slice bound $$Q^{\perp }$$ , their iterated by -mutations Z|_{n-1} }Q^{\perp .

The aim of this paper is to explain the bialgebra structure of the positive part of the quantized universal enveloping algebra (Drinfeld-Jimbo quantum group) of type Anusing the Lie algebra theory concepts. Recently has been introduced a generalization of Lie algebras, the basic T -Lie algebras [1]. Using the T -Lie algebra concept some new (we think) quantum groups of type Ancan be constructed...

Recent interests in quantum groups are stimulated by their marvelous relations with quantum Yang-Baxter equations, conformal field theory, invariants of links and knots, and q-hypergeometric series. Besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q-deformations or quantum analogues of known st...

We prove an analog of Janson’s strong hypercontractivity inequality in a class of non-commutative “holomorphic” algebras. Our setting is the q-Gaussian algebras Γq associated to the q-Fock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq , a q-Segal-Bargmann transform, and prove Janson’s strong hypercontractivity L(Hq)→ L(Hq) for r an even integer.