In mathematicsa free Lie algebraover a given field Kis a Lie algebra generated by a set Xwithout any imposed relations other than the defining relations of alternating bilinearity and the Jacobi identity. In the language of category theory, the functor sending a set X to the Lie algebra generated by X is the free functor from the category of sets to the category of Lie algebras.
That is, it is left adjoint to the forgetful functor. The free Lie algebra on a set X is naturally graded. The 0-graded component of the free Lie algebra is just the free vector space on that set.
One can alternatively define a free Lie algebra on a vector space V as left adjoint to the forgetful functor from Lie algebras over a field K to vector spaces over the field K — forgetting the Lie algebra structure, but remembering the vector space structure. The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X. This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.
Free Lie algebra
Ernst Witt showed that the number of basic commutators of degree k in the free Lie algebra on an m -element set is given by the necklace polynomial :. The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra. An explicit basis of the free Lie algebra can be given in terms of a Hall setwhich is a particular kind of subset inside the free magma on X.
Elements of the free magma are binary treeswith their leaves labelled by elements of X. Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory due to Philip Hall and Witt. In particular there is a basis of the free Lie algebra corresponding to Lyndon wordscalled the Lyndon basisso named after Roger Lyndon.
This is also called the Chen—Fox—Lyndon basis or the Lyndon—Shirshov basis, and is essentially the same as the Shirshov basis. Serre's theorem on a semisimple Lie algebra uses a free Lie algebra to construct a semisimple algebra out of generators and relations. The Milnor invariants of a link group are related to the free Lie algebra on the components of the linkas discussed in that article.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. This is equivalent to being the unique minimum word in lexicographic order among all its rotations. How to understand the linear dependence relation between general right-associative Lie monomials and what makes the Lyndon words the indexing set for the basis of free Lie algebras?
How the rotation of words affects the Lyndon words. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Lyndon basis of free Lie algebras Ask Question. Asked 8 months ago. Active 8 months ago. Viewed times.
Kindly share your thoughts. Thank you. GA GA 4 4 silver badges 16 16 bronze badges. It was an absolute pleasure to read, and I'm certain that the book has a good discussion of why there is a basis in correspondence with Lyndon words. I have edited the question, kindly reconsider this. I shall check this book. I did it with SageMath and github.
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Full text of " Lyndon-Shirshov basis and anti-commutative algebras " See other formats Lyndon-Shirshov basis and anti-commutative algebras" L. In this paper we give another approach to definition of Lyndon-Shirshov basis, i. Hall . He proved that P. Hall long commutators  form a linear basis of the algebra. He used a deg-ordering of monomials - monomial of greater degree is greater.
In what follow "monomial" would mean "non-associative monomial" "non-associative word" in Kurosh's terminology , adopted by Shirshov, see . Any deg-ordering of course satisfies the above condition.
Now this series of bases of Lie X is called Hall-Shirshov or even Hall bases the later is used in . In  1 the first author found another ordering of non-associative monomials with Shirshov's condition and as a result he found a linear basis of Lie X compatible with the lower central series of Lie X the result was rediscovered by C. Reutenauer, see his book , Ch. It was a breakthrough in the subject that Shirshov and Chen-Fox-Lyndon in the same year!
If one starts to construct the Hall-Shirshov basis of Lie X using the lex-ordering then one will get Lyndon- Shirshov basis automatically. There is a one to one correspondence between Lyndon-Shirshov monomials and Lyndon-Shirshov words.
At last [a[a[a6]]] is a Lyndon-Shirshov Lie monomial. Lyndon-Shirshov basis became one of popular bases of free Lie algebras cf. One of the main applications of Lyndon-Shirshov basis is the Shirshov's theory of Grobner-Shirshov bases theory for Lie algebras . Original Shirshov  definition is as follows. At last Shirshov said to the first author: "Your result pushes me to publish my result".
It is easy to see that regular monomials are defined inductively starting with Xi. From l - 3 it easily follows that w is the longest proper regular suffix of u, see .Algebraic number theory. Noncommutative algebraic geometry. Lie algebras are closely related to Lie groupswhich are groups that are also smooth manifolds : any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings Lie's third theorem.
This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras tangent vectors near the identity may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the s,  and independently discovered by Wilhelm Killing  in the s. The name Lie algebra was given by Hermann Weyl in the s; in older texts, the term infinitesimal group is used.
The dimension of a Lie algebra is its dimension as a vector space over F. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension. See the classification of low-dimensional real Lie algebras for other small examples. However, it is flexible.
Nonetheless, much of the terminology of associative rings and algebras is commonly applied to Lie algebras. In fact, because every one dimensional sub-vector space of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal.
For any simple lie algebra, all abelian Lie algebras can never be ideals. See also semidirect sum of Lie algebras.
Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra Levi subalgebra. This is a derivation as a consequence of the Jacobi identity. The outer derivations are derivations which do not come from the adjoint representation of the Lie algebra. A split real form of a complex semisimple Lie algebra cf. Real form and complexification is an example of a split real Lie algebra.
See also split Lie algebra for further information. Such Lie algebras are called abeliancf. Any one-dimensional Lie algebra over a field is abelian, by the alternating property of the Lie bracket. The following are examples of Lie algebras of matrix Lie groups: . A representation is said to be faithful if its kernel is zero. Ado's theorem  states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space. One important aspect of the study of Lie algebras especially semisimple Lie algebras is the study of their representations.
Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.
Indeed, in the semisimple case, the adjoint representation is already faithful. In the semisimple case over a field of characteristic zero, Weyl's theorem  says that every finite-dimensional representation is a direct sum of irreducible representations those with no nontrivial invariant subspaces.
The irreducible representations, in turn, are classified by a theorem of the highest weight. The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations.Algebraic number theory. Noncommutative algebraic geometry. In mathematicsan associative algebra is an algebraic structure with compatible operations of addition, multiplication assumed to be associativeand a scalar multiplication by elements in some field.
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 vector space over K. In this article we will also use the term K -algebra to mean an associative algebra over the field K. A standard first example of a K -algebra is a ring of square matrices over a field Kwith the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring.
In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring Rinstead of a field: An R -algebra is an R -module with an associative R -bilinear binary operation, which also contains a multiplicative identity.
For examples of this concept, if S is any ring with center Cthen S is an associative C -algebra. Let R be a fixed commutative ring so R could be a field. An associative R -algebra or more simply, an R -algebra is an additive abelian group A which has the structure of both a ring and an R -module in such a way that the scalar multiplication satisfies.
Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that. If A itself is commutative as a ring then it is called a commutative R -algebra. The definition is equivalent to saying that a unital associative R -algebra is a monoid object in R -Mod the monoidal category of R -modules. By definition, a ring is a monoid object in the category of abelian groups ; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A.
For example, the associativity can be expressed as follows. By the universal property of a tensor product of modulesthe multiplication the R -bilinear map corresponds to a unique R -linear map. An associative algebra amounts to a ring homomorphism whose image lies in the center.
The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R.
How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: generic matrix ring. A homomorphism between two R -algebras is an R -linear ring homomorphism. The class of all R -algebras together with algebra homomorphisms between them form a categorysometimes denoted R -Alg. The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center.
In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics. Let A be an algebra over a commutative ring R. Let A be a finite-dimensional algebra over a field k.
Then A is an Artinian ring. As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field k. Now, a reduced Artinian local ring is a field and thus the following are equivalent . Since a simple Artinian ring is a full matrix ring over a division ring, if A is a simple algebra, then A is a full matrix algebra over a division algebra D over k ; i.
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