In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle productX ⧢ Y of two words X, Y: the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation.
The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set.
Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words.
The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative.
mathematics, a shufflealgebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of...
Riffle shuffles may be used to define the shufflealgebra. This is a Hopf algebra where the basis is a set of words, and the product is the shuffle product...
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the...
that the Lyndon words are algebraically independent elements of this shufflealgebra, and generate it; thus, the shufflealgebra is isomorphic to a polynomial...
enveloping algebra of a free Lie algebra on a finite set is the shufflealgebra. This essentially follows because universal enveloping algebras have the...
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle...
began what would become a fruitful collaboration on the interplay between algebra and topology. In 1944 and 1945, he directed Columbia University's Applied...
who co-founded category theory (with Saunders Mac Lane) and homological algebra. He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent...
algebra (also known as a Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is...
be extended to the entire algebra. Examining the lifting in detail, one observes that the coproduct behaves as the shuffle product, essentially because...
In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the...
Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though...
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the...
This article uses algebraic notation to describe chess moves. Fischer random chess, also known as Chess960 ('chess nine-sixty'), is a variation of the...
Westerland, Craig (2023-03-05). "Fox-Neuwirth-Fuks cells, quantum shufflealgebras, and Malle's conjecture for function fields". arXiv:1701.04541 [math...
{\displaystyle (p,q)} shuffles. Loday 2001, p. 45 Dzhumadil'daev, A.S.; Tulenbaev, K.M. (2005). "Nilpotency of Zinbiel algebras". J. Dyn. Control Syst...
expression. Finally the Nichols algebra is then W n := ∑ σ ∈ S n s ( σ ) : V ⊗ n → V ⊗ n (quantum symmetrizer or quantum shuffle map) {\displaystyle {\mathfrak...
elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic...
The use of this model is beneficial only when the optimized distributed shuffle operation (which reduces network communication cost) and fault tolerance...
are Lie algebras, and S : V → V and T : W → W are Lie algebra homomorphisms, the Kronecker sum of A and B represents the induced Lie algebra homomorphisms...
in numerous programming languages, in spreadsheets, and in relational algebra. The listagg function, as defined in the SQL:2016 standard aggregates data...
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