# Realization of Lie algebras and differential calculus on noncummutative spaces

Martinić, Tea (2016) Realization of Lie algebras and differential calculus on noncummutative spaces. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In the first part of the thesis we study extensions of a Lie algebra $$\mathfrak{g}_0$$ with basis $$\{X1, X2, \dots, Xn\}$$ by an Abelian familiy of generators $$T_{\mu \nu}, 1 \leq \mu,\nu \leq n$$, which act on the enveloping algebra $$U(\mathfrak{g}_0)$$. This action describes the commutation relations between $$X_{\mu}$$ and arbitrary monomials in $$U(\mathfrak{g}_0)$$. Furthermore, we study realizations of the Lie algebra $$\mathfrak{g}_0$$, i.e. embedding of $$\mathfrak{g}_0$$ into $$\hat{A}_n$$ where $$\hat{A}_n$$ is the completion of the $$n$$-th Weyl algebra $$A_n$$ by the degree of the differential operator $$\partial^{k_1}_1 \partial^{k_2}_2 \dots \partial^{k_n}_n$$. In particular, we study realizations which induce the symmetric ordering on the enveloping algebra $$U(\mathfrak{g}_0)$$. Using properties of the extended Lie algebra it is shown that the corresponding symmetric realization is given in terms of the generating function for the Bernoulli numbers. To each realization of the Lie algebra $$\mathfrak{g}_0$$ one can associate a star–product on the symmetric algebra $$X=\mathbb{C}[x_1, x_2, \dots , x_n] \subset A_n$$ which is defined using the canonical action of the algebra $$A_n$$ on the subalgebra $$X$$. We introduce the notion of left–right dual star–products and the corresponding realizations. The left–right duality is studied in detail in case of the symmetric realization using the aforementioned extension of the Lie algebra $$\mathfrak{g}_0$$. The second part of the thesis deals with bicovariant differential calculus on the quantum space $$U(\mathfrak{g}_0)$$. For this purpose we construct a Lie superalgebra $$\mathfrak{g}=\mathfrak{g}_0 \bigoplus \mathfrak{g}_1$$ where the basis elements $$\xi_1, \xi_2, \dots , \xi_m$$ of the odd part $$\mathfrak{g}_1$$ are interpreted as one–forms on the space $$U(\mathfrak{g}_0)$$. By generalizing the result from the first part of the thesis we obtain an extension of $$\mathfrak{g}$$ by an Abelian family of generators $$T_{\mu\nu}$$ whose action on the enveloping algebra $$U(\mathfrak{g})$$ describes the commutation relations between one–forms and monomials in $$U(\mathfrak{g}_0)$$. We also construct a realization, i.e. an embedding of the superalgebra $$\mathfrak{g}$$ into a completion of the Clifford–Weyl algebra $$\hat{A}_{n,m}$$. In case when $$dim(\mathfrak{g}_0) = dim(\mathfrak{g}_1)$$ we define an exterior derivative $$d: U(\mathfrak{g}_0) \to \Omega$$ where $$\Omega=\bigoplus^n_{\mu=1} U(\mathfrak{g}_0)\xi_\mu$$ is an $$U(\mathfrak{g}_0)$$–bimodule. The first order differential calculus $$(d, \Omega)$$ is bicovariant with respect to the primitive Hopf structure of $$U(\mathfrak{g}_0)$$. Using the realization of the Lie superalgebra $$\mathfrak{g}$$, the differential calculus is obtained as a deformation of the classical differential calculus on the Euclidean space.