We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a...

Since the invention of canonical bases of quantized enveloping algebras, one of the main problems has been to establish what they look like. Explicit formulas are only known in a few cases corresponding to root systems of low rank, namely A1 (trivial), A2 ([Lusztig 90]), A3 ([Xi 99a]), and B2 ([Xi 99b]). Furthermore, there is evidence suggesting that for higher ranks the formulas become so comp...

The algebra Sn of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting, left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra Sn is a noncommutative, non-Noetherian algebra of classical Krull dimension 2n and of global dimension n and is not a domain. If the ground field K has characteristic zero then the alg...

The Lie algebra W = DerA is called the Witt algebra. It consists of “vector fields” f∂, f ∈ A. In particular, dimF W = dimF A = p. As any Lie algebra of derivations of a commutative algebra over F, W has a canonical structure of a restricted Lie algebra. Recall that a restricted Lie algebra is a Lie algebra over F with an additional unary (in general, non-linear) operation g 7→ g satisfying the...

A lot of recent activity has been directed toward various constructions of “natural” bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig’s construction of a canonical basis, and the pioneering construction of the Kazhdan–Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig’s Lemma that we apply to a...

For symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the modified quantized enveloping algebra U̇(g) and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra g we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of U̇(g), where the root category is the T -orbit category ...

In 7] a hierarchy of directed acyclic graph (dag) algebras is deened. The dags belonging to an algebra are deened up to an isomorphism which can be expressed by a congruence over a free term algebra, i.e. a dag is described by a congruence class of terms. In this paper we deal with the following problem: can be deened a canonical form in every congruence class and a canonical (connuent and term...

It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, deened by means of axioms similar to Woro-nowicz's. This gives rise to Lie algebra-like generators and relations for the locally nite part of the quantised enveloping algebra, and suggests a canonical Poincar e-Birkhoo-Witt basis.

We introduce relativized modal algebra homomorphisms and show that the category of modal algebras and relativized modal algebra homomorphisms is dually equivalent to the category of modal spaces and partial continuous p-morphisms, thus extending the standard duality between the category of modal algebras and modal algebra homomorphisms and the category of modal spaces and continuous p-morphisms...