Related papers: Nilpotent Lie Groups in Clifford Analysis: Five Di…
Albuquerque and Majid have shown how to view Clifford algebras $\cl_{p,q}$ as twisted group rings whereas Chernov has observed that Clifford algebras can be viewed as images of group algebras of certain 2-groups modulo an ideal generated by…
We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new…
We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth…
The n-qubit Pauli group and its normalizer the n-qubit Clifford group have applications in quantum error correction and device characterization. Recent applications have made use of the representation theory of the Clifford group. We apply…
This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the…
We introduce the symplectic group $\mathrm{Sp}_2(G, \sigma)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $\sigma$. Our construction recovers several classical Lie…
In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible…
Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of four dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear…
We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014.…
We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular,…
We survey a few results on the boundedness of operators arising from the Weyl-Pedersen calculus associated with irreducible representations of nilpotent Lie groups.
This is a survey article with a limited list of references (as required by the publisher) which appears in the Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol.4, pp.94--104.
We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a…
We formulate a Boolean algebra in the set of idempotents of Clifford algebra Cl($R^{n,n}$) and within this frame we examine different formulations of the Boolean Satisfiability Problem in Clifford algebra. Exploiting the isomorphism between…
An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centred around the simultaneous null…
We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of…
We present a Riesz-like hyperholomorphic functional calculus for a set of non-commuting operators based on the Clifford analysis. Applications to the quantum field theory are described. Keywords: Functional calculus, Weyl calculus, Riesz…
In the present paper, we determine the group of automorphisms of pseudo $H$-type Lie algebras, which are two-step nilpotent Lie algebras closely related to the Clifford algebras $\Cl(\mathbb R^{r,s})$.
The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras…
Left-invariant optimal control problems on Lie groups form an important class of problems with big symmetry group. They are interesting from the theoretical point of view since they often can be completely studied, and general features can…