Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

Given a function and a vector field X defined on M, the Lie derivative of a function ƒ along a vector field is simply the application of the vector field.

* Given the action of a Lie algebra g on a manifold M, the set of g-invariant vector fields on M is a Lie algebroid over the space of orbits of the action.

* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.

Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).

Given any compact Lie group G one can take its identity component G < sub > 0 </ sub >, which is connected.

Given a basis e < sup > i </ sup > of the Lie algebra g, the matrix elements of the Killing form are given by

Given an element x of a Lie algebra, one defines the adjoint action of x on as the endomorphism with

Given a group G, a factor group G / N is abelian if and only if ≤ N.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given our formula φ, we group strings of quantifiers of one kind together in blocks:

Given that different groups in society have different beliefs, priorities, and interests, to which group would the media tailor its bias?

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

* Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given a series with values in a normed abelian group G and a permutation σ of the natural numbers, one builds a new series, said to be a rearrangement of the original series.

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

Given that France and Britain had been at war since early 1793, administering or making such oaths turned the society into something more than a liberal pressure group.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given that and record company pressure to record more accessible, radio-friendly material similar to their first album – something Lee, Lifeson and Peart were unwilling to do – the trio feared that the end of the group was near.

Given any group G, the group consisting of only the identity element is a trivial group and being a subgroup of G is called the trivial subgroup of G.

Given these orbital elements and the physical characteristics known so far, Ananke is thought to be the largest remnant of an original break-up forming the Ananke group.

Given the stage name Jett Powers by top Hollywood agents Gabey, Lutz, Heller and Loeb, he took acting and singing lessons, and appeared in movies with small roles.

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

Given a manifold Q, a vector field X on Q ( or equivalently, a section of the tangent bundle TQ ) can be thought of as a function acting on the cotangent bundle, by the duality between the tangent and cotangent spaces.

Given the suspension of semicha, any beth din existing in medieval or modern times is in theory a court of laymen, acting as arbitrators.

Given the physical resemblance between the Eternals and Olympians, both parties formed an alliance, with the Eternals acting representing the gods on Earth.

Given this interpretation of morality, it is not possible for the egoist to do better by acting against morality.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

Given that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given its heavy dependence on imported energy, Japan has aimed to diversify its sources.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).