|
Partition Algebra
The partition algebra is an associative algebra with a basis of partition of a set, set-partition diagrams and multiplication given by diagram Path (topology)#Path composition, concatenation. Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley-Lieb algebra, Temperley–Lieb algebra, or the group ring, group algebra of the symmetric group. Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group. Definition Diagrams A partition of 2k elements labelled 1,\bar 1, 2,\bar 2,\dots, k,\bar k is represented as a diagram, with lines connecting elements in the same subset. In the following example, the subset \ gives rise to the lines \bar 1 - \bar 4, \bar 4 -\bar 5, \bar 5 - 6, and could equivalently be represented by the lines \bar 1- 6, \bar 4 - 6, \bar 5 - 6, \bar 1 - \bar 5 (for instance). For n\in \mathbb and k\in \mathbb^*, the partition algebra P_k(n) is defined by a \mathbb-basis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a module or vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a commutative ring ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
