We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that are normalized by the sl$_2$-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_n^{(-\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The connection goes both ways---we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.

We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type $C(n)$, $D(m,n)$, $D(2,1;\alpha)$ $(\alpha \in \mathbb{F} \setminus \{0,-1\})$, $F(4)$, and $G(3)$.

We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.

We study the representations of extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$-th primitive root of unity ($\mathbb{C}_q$ is the quantum torus in two variables). We first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a suitable number of copies is a quotient of $s\ell_{\ell+1} (\mathbb{C}_q)$. Thus any finite dimensional irreducible module for $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of $s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$ comes from above. We then construct modules for the extended affine Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1 \oplus \mathbb{C} d_2$ which is integrable and has finite dimensional weight spaces.

We show that an $\RA$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\RA$ loop can be normal in its unit loop. Over fields, this can never happen.

We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\bbC$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

An integral formula is derived for the spherical functions on the symmetric space $G/K=\break \SO_0(p,q)/\SO(p)\times \SO(q)$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra $\a$ of the abelian part in the decomposition $G=KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel formula. In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

A theorem of Kor\'anyi and Wolf displays any Hermitian symmetric domain as a Siegel domain of the third kind over any of its boundary components. In this paper we give a simple proof that an analogous realization holds for any self-adjoint homogeneous cone.

It is shown that every simple complex Lie algebra $\fg$ admits a 1-parameter family $\fg_q$ of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for $U_q(\fg)$-modules; here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$. From this it follows that the multiplication on $\fg_q$ is $U_q(\fg)$-invariant. In the special case $\fg = {\ss}(2)$, the structure constants for the deformation ${\ss}(2)_q$ are obtained from the quantum Clebsch-Gordan formula applied to $V(2)_q \otimes V(2)_q$; here $V(2)_q$ is the simple 3-dimensional $U_q\bigl({\ss}(2)\bigr)$-module of highest weight $q^2$.

For every field $F$ of characteristic $p\geq 0$, we construct an example of a finite dimensional nilpotent $F$-algebra $R$ whose adjoint group $A(R)$ is not centre-by-metabelian, in spite of the fact that $R$ is Lie centre-by-metabelian and satisfies the identities $x^{2p}=0$ when $p>2$ and $x^8=0$ when $p=2$. The existence of such algebras answers a question raised by A.~E.~Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4.