This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave for short) if a: 2 a, ,a ,+ , for all 1 5 i 5 n 1. Clearly a log-concave sequence of positive terms is unimodal. Let us say that the sequence a,, a,, . . . , a, has no internal zeros if there do not exist integers 0 5 i < j < k 5 n satisfying a, f 0, a, = 0, ak # 0. Then in fact a nonnegative log-concave sequence with no internal zeros is unimodal. The sequence a,, a , , . . . , a, is called symmetric if a, = a,-, for 0 5 i 5 n. Thus a symmetric unimodal sequence a,, a, , . . . , a,, has its maximum at the middle term ( n even) or middle two terms (n odd). We also say that a polynomial a, + a,4 +. . + an$ has a certain property (such as unimodal, log-concave, or symmetric) if its sequence a,, a,, . . . , a, of coefficients has that property. Our object here is to survey the surprisingly rich variety of methods for showing that a sequence is log-concave or unimodal. For each method we will give examples of its applicability to combinatorially defined sequences that arise naturally from problems in algebra, combinatorics, and geometry. We make no attempt, however, to give a comprehensive account of all work done in this area.

Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry

Let G be a connected linear algebraic group, and p a rational representation of G on a finite-dimensional vector space V , all defined over the complex number field C . We call such a triplet ( G, p, V ) a prehomogeneous vector space if V has a Zariski-dense G -orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces when p is irreducible, and to investigate their relative invariants and the regularity.

A classification of irreducible prehomogeneous vector spaces and their relative invariants

Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Yangians and Classical Lie Algebras

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/28397/0000172.pdf;sequence=1

Nonintersecting paths, pfaffians, and plane partitions

Renormalization theory in quantum gravity, among other applications, continues to stimulate many attempts to calculate asymptotic expansions of heat kernels and other Green functions of differential operators. Computer algebra systems now make it possible to carry these calculations to high orders, where the number of terms is very large. To be understandable and usable, the result of the calculation must be put into a standard form; because of the subtleties of tensor symmetry, to specify a basis set of independent terms is a non-trivial problem. This problem can be solved by applying some representation theory of the symmetric, general linear and orthogonal groups. In this work the authors treat the case of scalars or tensors formed from the Riemann tensor (of a torsionless, metric-compatible connection) by covariant differentiation, multiplication and contraction. (The same methods may be applied readily to other tensors.) The authors have determined the number of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric, and exhibited a basis for these invariants up through order eight. For tensors of higher rank, they present bases through order six; in that case some effort is required to match the familiar classical tensor expressions (usually supporting reducible representations) against the lists of irreducible representations provided by the more abstract group theory. Finally, the analysis yields (more easily for scalars than for tensors) an understanding of linear dependences in low dimensions among otherwise distinct tensors.

https://iopscience.iop.org/article/10.1088/0264-9381/9/5/003/pdf

Normal forms for tensor polynomials. I. The Riemann tensor

Tensor and spinor methods are used to derive branching rule formulae for the embedding of one classical Lie group in another. These formulae involve operations on S-functions. By the judicious use of identities satisfied by certain infinite series of S-functions, they are reduced to forms which may be used very efficiently. Eleven sets of the branching rule formulae derived are as simple as possible, in that they involve only a sum of positive terms, whilst four other sets involve some negative terms which ultimately cancel. The advantage of using a composite notation, both for mixed tensor and for spinor representations, is made apparent. A comparison is made with methods used to derive branching rules based on mapping from one weight space to another.

https://iopscience.iop.org/article/10.1088/0305-4470/8/4/004/pdf

Branching rules for classical Lie groups using tensor and spinor methods

Modification rules, expressible in terms of the removal of continuous boundary hooks, are derived which relate nonstandard irreducible representations (IR's) of the unitary, orthogonal, and symplectic groups in n dimensions to standard IR's. Tensorial methods are used to derive procedures for reducing the outer products of IR's of U(n), O(n), and Sp(n), and for reducing general IR's of U(n) specified by composite Young tableaux with respect to the subgroups O(n) and Sp(n). In these derivations the conjugacy relationship between the orthogonal and the symplectic groups is fully exploited. The results taken in conjunction with the modification rules are valid for all n.

Modification Rules and Products of Irreducible Representations of the Unitary, Orthogonal, and Symplectic Groups

Kac distinguished between typical and atypical finite‐dimensional irreducible representations of the Lie superalgebras sl(m/n) and provided an explicit character formula appropriate to all the typical representations. Here, the range of validity of some character formulas for atypical representations that have been proposed are discussed. Several of them are of the Kac–Weyl type, but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl(3/4). Having ruled out, therefore, all such formulas, a completely new extension of the Kac–Weyl character formula is proposed. The validity of this formula in the case of all covariant tensor irreducible representations is proved, and some evidence in support of the conjecture that it covers all irreducible representations of sl(m/n) is presented.

Character formulas for irreducible modules of the Lie superalgebras sl(m/n)

A review is given of the application of S-function techniques to the evaluation of Kronecker products of irreducible representations of compact semisimple Lie groups. Explicit formulae are derived for all irreducible representations of all such groups. Recent developments involving composite Young diagrams are brought to fruition and the vexed problem of SO2k is dealt with completely. New branching rules for the classical groups are given in an appendix. These are exploited in the evaluation of Kronecker products by means of a technique which is applied to both classical and exceptional groups. A discussion is made of various modification rules which are needed to express the final results in standard form.

https://iopscience.iop.org/article/10.1088/0305-4470/16/8/006/pdf

Ronald C. King

Papers

Normal forms for tensor polynomials. I. The Riemann tensor

Branching rules for classical Lie groups using tensor and spinor methods

Modification Rules and Products of Irreducible Representations of the Unitary, Orthogonal, and Symplectic Groups

Character formulas for irreducible modules of the Lie superalgebras sl(m/n)

Kronecker products for compact semisimple Lie groups